I’m starting this thread in ‘interactive seminar form’ for the benefit of beginners and burgeoning chemists who haven’t had the benefit of a higher education in science and want to understand better what causes chemical bonds to form.

Being a normal thread anyone is welcome (and it’s FREE! ) to contribute, correct me or ask questions. Of more knowledgeable contributors I ask only to take one thing into account: this thread is not about cutting edge Quantum Physics, rather because of its stated intent shouldn’t really exceed A-level/1<sup>st</sup> year Uni levels of sophistication. If interest warrants it, we can always build up.

To the extent of the possible and practical I will only move on from one concept to the next if I feel most ‘students’ have reached sufficient understanding of each concept.

It’s a two part ‘course’:

Part I:

Part II:

First instalment tomorrow.

[Edited on 9-7-2015 by blogfast25]

- Wave-particle duality

- Complex numbers

- Wave functions

- Time-independent Schrödinger equation

- Introduction to quantum systems: the particle-in-a-box model (one dimension)

- Probability of locating particle at a given position and the standing wave analogy

- Quantization of energy

- Alternate method of determining energy levels

- Emission spectrum of particle in a box

- Normalization of wave functions

- Wave function orthogonality

- Parity of wave functions

- Corrections regarding probability and probability densities

- The Quantum Harmonic Oscillator

- Operators: Classic and Quantum

- The Uncertainty Principle

- Another math mini-excursion: Cartesian and Spherical Coordinates - The Hydrogen Atom’s Schrodinger Equation

- Hydrogen Atom Wave Functions - Quantum numbers and Orbital naming

- Hydrogen Energy Levels

- Multi-electronic elements: Helium and beyond - The Pauli Exclusion Principle (PEP)

- The Aufbau Principle

- The Mother of all Covalent Bonds: the σ(ss) bond in Molecular Dihydrogen

- Bonding and anti-bonding MOs between p-orbitals and between s and p-orbitals

- Double bonds and triple bonds - Conjugated double bonds and Aromaticity

- Molecular shapes and molecular orbital repulsion

- Hybridisation - Water and the oxonium ion

- Higher Hybridisations - A partial explanation of the colours of transition cation complexes

- Electronegativity - Polar diatomic molecules

- Polar and non-polar molecules

- Bond Enthalpies

- Gerate and Ungerate orbitals: bonding/anti-bonding in σ and π bonds

- When a Covalent Bond is not a Covalent Bond Anymore

- Born-Haber Thermochemical cycle: Ionisation Energy, Electron Affinity and Lattice Energy

- Electrophilic Addition Reaction Mechanism

- A glimpse into the reaction mechanism of the synthesis of Indole-3-acetic acid

- The Quantum Theory of Hydrogen Bonding

- Lewis acid base theory (digest) and an example of OC catalysis explained

- Structural notation/Fisher esterification reaction mecanism

**Additional Contributions**

- Small de Broglie wavelength of macroscopic objects and its implications

- History of black-body radiation

- Classical behavior of macroscopic objects a result of decoherence not size

- Decoherence and the correspondence principle

- Operators, probability distributions, and the Schrödinger equation

- Spectral lines and energy levels

- Definite eigenstates and linear combinations of eigenstates

- Superposition, wave function collapse, and the Heisenberg uncertainty principle

- Recap on the Hamiltonian

- Deprotonation of carboxylic acids and alcohols

- More resonance structures

[Edited on 31-10-2015 by Bert] - Small de Broglie wavelength of macroscopic objects and its implications

smaerd - 8-7-2015 at 17:25

Cool idea blogfast25. Q&M was one of my favorite topics as an undergraduate. So I'll be really happy to read through this as it grows. I found the atkins physical chemstry book to be exceptional for teaching Q&M from a chemists perspective. At least the entry level models. Also the book, "physical chemistry a biochemists perspective" was really great at being simple.

Edit - I may even have a primer on some of the calculus involved. I was writing it for an applied mathematics book. Hmm...

[Edited on 9-7-2015 by smaerd]

Zombie - 8-7-2015 at 19:20

It's already a "Cliff Hanger". I have to put my popcorn up for tomorrow.

Thank you for doing this Blog. I'm pretty excited to see what I actually know, can grasp, and can learn.

blogfast25 - 9-7-2015 at 07:19

Thanks Zombie man.

Smaerd, there's not going to be a whole lot of calculus rocking here but add some if you please (within reason, of course)

And we're off!

At the level of the very small (atoms and smaller) things get decidedly strange because Classical Newtonian Laws governing moving objects no longer apply. A number of unexpected experimental results around the turn of the 19th Century showed that waves could behave like particles and vice versa (the list below is far from exclusive).

1. The photo electric effect:

http://hyperphysics.phy-astr.gsu.edu/hbase/mod1.html#c2

2. Compton Scattering:

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/comptint....

3. Davisson-Germer Experiment:

http://hyperphysics.phy-astr.gsu.edu/hbase/davger.html#c1

While 1. and 2. show electromagnetic waves behaving like particles, 3. shows moving electrons behaving like waves!

The Davisson-Germer Experiment served as an experimental verification of the DeBroglie Hypothesis:

http://hyperphysics.phy-astr.gsu.edu/hbase/debrog.html#c1

A more formal treatment of the DeBroglie Hypothesis and Wavelength can be found here:

http://hyperphysics.phy-astr.gsu.edu/hbase/debrog.html#c3

Free-moving microscopic (atomic or smaller) particles, like electrons, with momentum p (= mv, m = mass in kg, v = speed in m/s) thus behave like waves with wavelength:

[Edited on 9-7-2015 by blogfast25]

aga - 9-7-2015 at 09:28

Nice one !

Thanks for taking the time to put this together bloggers.

Can us eejits ask questions yet ?

Plancks Constant (https://en.wikipedia.org/wiki/Planck_constant) being the smallest amount by which a system must change ?

blogfast25 - 9-7-2015 at 10:06

Yes, questions/comments are now taken, before moving on to the next session.

Aga, for NOW it's best to look at

But

To ALL 'students': bear in mind the profound differences between moving particles and classic waves, to appreciate just how momentous the discoveries described above really were. They were the start of description of the 'weirdness' of the QM world, known as Quantum Physics.

For fundamental differences between waves and moving objects, see:

http://hyperphysics.phy-astr.gsu.edu/hbase/mod1.html#c1 and SCROLL DOWN to:

[Edited on 9-7-2015 by blogfast25]

smaerd - 9-7-2015 at 11:14

@Aga- the photoelectric effect can be used to show how the value of plancks constant can be experimentally determined, and sort of what it means in at least that scenario. (http://physicsnet.co.uk/a-level-physics-as-a2/electromagneti...)

The DeBroglie wave idea is one of my favorite things that came at the dawn of Q&M . I had a physics professor who posed a very fun question.

Question: Say a man who took first year physics ends up in court for shooting a man (who survived). He claims that he had no idea that his bullet would actually hit the stationary man. While a bystander took a video on his camera phone which shows he aimed right at the guy! He further claimed that the bullet can be considered as a wave and it's wave-like nature gave an uncertainty in his aim that was significant enough to state that he had reason to believe he would not hit the man.

The cops found that his gun fired a bullet at 1000m/s, and the average bullet mass was 3grams (0.003kg). Is what the shooter claims a valid defense? (Hint: use an equation from this page- http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/debrog2.h...). Assume the bullet traverses a 1-D path and ignore relativistic effects of the bullet (ex: just use the rest mass).

Answer here: The debroglie wavelength of the bullet can be calculated as follows. First we know that the 'particle' is not travelling at the speed of light, so substitute c(speed of light) for v(velocity), this is one of debroglies postulates iirc. λ = hv/pv => h/p => h/(m*v)

= 6.63*10^(-34)J*s/(3.0kg*m/S) = 2.21 *10^(-34) m

So no, the man cannot make the claim he did. The object is essentially newtonian! Another way to approach the problem would be the Hiesenberg Uncertainty principle. Hope I did this right,

aga - 9-7-2015 at 12:35

Whoosh ! Went right over my head by introducing too much too fast.

Wading through the question the best i can, am i to assume that if i could calculate the wavelength (got NaN a lot on the calculator linked to) of the bullet as if it were a wave then if the cycle length of that wave were such that it could not impart energy to the target then it'd all be OK ?

Bullet or Wave : it'd still hit the target if aimed correctly.

blogfast25 - 9-7-2015 at 13:01

Aga: I'm not sure where he's going with this either but try and calculate the DeBroglie wavelength of the bullet first. Then we'll see... (Hint: it's extremely small!)

No calculator needed even: p = 0.003 kg x 1000 m/s = 3 kg m/s,

So λ =

[Edited on 9-7-2015 by blogfast25]

aga - 9-7-2015 at 13:10

Hmm.

With this jobbie λ = h/p

we got h/(0.003 * 1000), so

6.62606957 x 10<sup>-34 </sup>/ 3

2.2086898 x 10<sup>-34</sup>

= 0.00000000000000000000022086898 pico metres

So very high energy indeed. Hang the man.

[Edited on 9-7-2015 by aga]

blogfast25 - 9-7-2015 at 13:22

Quote: Originally posted by aga |

No, not high energy. The energy is 'classical': E = 1/2 mv<sup>2</sup> = 150 kJ, enough to kill man of course.

But the bullet's DeBroglie wavelength is extremely small and shot accuracy isn't remotely affected by it. The bullet behaves 'Classically'.

[Edited on 9-7-2015 by blogfast25]

aga - 9-7-2015 at 13:31

I mean 'very high' in terms of affecting human tissues, not 'high' in comparison with the output of a Star.

Why would a longer wavelength affect the accuracy ?

I can see how the amplitude could be seern as a way for the wave to 'miss' the target but not the wavelength.

blogfast25 - 9-7-2015 at 13:42

Quote: Originally posted by aga |

It's to do with the Uncertainty Principle, which I will hit on a few sessions down the line. I'd rather not right now because I don't like mixing concepts in stuff like this (I know how confusing things can get). I will use this specific bullet example when I get to it.

Of course I can't stop smaerd from doing it anyway.

+++++++++

Go back a bit to the Davisson-Germer experiment:

http://hyperphysics.phy-astr.gsu.edu/hbase/davger.html#c1

Here electrons are accelerated to an energy of 54 eV ("electron volt"). If you work out the DeBroglie wavelength associated with those electrons, it turns out to be in the same order of magnitude as the distance (1.65 Angstrom - an old unit of length) between crystal planes in the Ni crystal lattice. This caused

From the electrons' energy, the crystal structure and DeBroglie's calculation the latter's hypothesis could be confirmed experimentally. Clearer, quantitative proof of these electrons' 'wavy character' would be hard to find. Without the latter the electrons would simply bounce (be 'reflected') off the nickel.

[Edited on 10-7-2015 by blogfast25]

aga - 9-7-2015 at 13:52

Right oh. Did seem a bit of a leap from square 1.

I'll ignore all that then, and as a bonus, i'll not mention

[Edited on 9-7-2015 by aga]

blogfast25 - 9-7-2015 at 14:27

Quote: Originally posted by aga |

Now, THAT you see, I find hard to believe!

Looks like it's going to be one-on-one tuition anyway. Offer 2 cans of beans for the price of one and they'll queue for miles. A free course on the finest scientific paradigm? Two brothers and their dog show up! P-R-I-O-R-I-T-I-E-S, people!

smaerd - 9-7-2015 at 14:41

Yea sorry if that was too much too fast. The idea is basically to show that "big" things do not behave like small particles. If the particle had a very tiny mass then the debroglie wavelength becomes relevant.

The question is kind of slanted. The idea is basically that if the bullets wavelength was say 2 meter in radius. Then maybe the shooter would not be accountable because if it behaved like a particle (to collide) it would be less likely to hit the man. Similarly the de broglie wavelength of semi-trucks/lorry can be calculated. I liked it because it shows that something 'small' and 'fast' on the human scale still isn't really relevant for quantum mechanics.

Type in the mass of an electron into that equation and it's pretty clear that electrons have wavelengths that are not negligable.

blogfast25 - 9-7-2015 at 14:43

Fair point, smaerd.

Zombie would be late for his own funeral, so I'll wait to call it a day on this modul

[Edited on 9-7-2015 by blogfast25]

aga - 9-7-2015 at 14:48

Quote: Originally posted by blogfast25 |

Honest : i won't say anything at all about that until after the course.

More people will turn up later.

There's Timezones involved in this global classroom, so some are asleep right now - not cos they find it boring, just that it's 4am where they are.

Bot0nist - 9-7-2015 at 15:17

I'm very interested in this subject and will be following this thread as it progresses. I don't yet have any questions, and certainly don't posses anything of use to add, but I am very appreciative for blogfast's and smaerd 's time and contributions. Thank you, wizards. Threads like these bring needed balance to the "don't say duck" and "joke threads." Please, don't let a lack of posters deter you. I believe a great many people will read and learn from it, even if they don't post.

blogfast25 - 9-7-2015 at 15:35

Time zones: someone is going to have to do something about that, sooner or later!

Thanks Bot0nist. Your subscription fee of $0.00 will not be deducted from your charge card.

Magpie - 9-7-2015 at 18:17

This is fascinating material and a review for all I have forgotten.

I found the explanation for the shape of the black body radiation curve vs temperature to be the most astounding. This explains how the color of a heated metal indicates its temperature. Was it de Broglie or was it Planck that discovered this?

Metacelsus - 9-7-2015 at 18:28

Planck.

https://en.wikipedia.org/wiki/Planck%27s_law

Molecular Manipulations - 9-7-2015 at 19:52

Ditto what

Thanks for doing this,

blargish - 9-7-2015 at 20:55

Great idea for a thread!

The bullet question somewhat confused me... It's hard to match these quantum mechanical effects to real-life macro analogies. What I guess would make sense is that if the bullet's de Broglie wavelength matched (or at least was on the same scale as), say, the width of the nozzle of the gun, diffractive effects would lead one to believe (on the macroscopic scale) that there is some uncertainty in whether or not the bullet actually hits the guy right in front of it. Not exactly sure how correct this is, but the way I usually think of it is that even though something big like a truck has some tiny de Broglie wavelength, it could never pass through a sufficiently small opening for diffractive effects to occur, and thus its wave character to be even noticeable, whist electrons and other subatomic particles are on a scale such that these wave-like effects do come into play.

Also, I was actually lucky enough to do some labs this year regarding the photoelectric effect, and using the stopping potential for various frequencies to calculate planck's constant Physics just blows my mind, and I can't wait to see where this thread goes!

blogfast25 - 10-7-2015 at 03:17

Re. matter waves of large objects, one way of looking at it is like this.

λ = h/mv

With mass m an indication of size of the object. Obviously in the limit of m to ∞, λ tends to 0 m. 0 m obviously means no waves. The larger the object, the less 'wavy' it behaves.

Since as Zombie's gone AWOL, I'll put up the next modulette now. Z. will get detention and extra tuition, if he needs it.

blogfast25 - 10-7-2015 at 03:25

Complex numbers play a big part in QM/WM (although much less in its chemical applications, so we’ll keep it to a bare minimum). But since as you’ll see the symbol “

At the heart of complex numbers lays the imaginary number

The ‘imaginary’ number i is not a Real Number because square roots of negative number have no (Real) roots. But the square of i, that is i<sup>2</sup> = - 1, IS real (but negative). Complex Numbers can thus be used in calculations, sometimes yielding other CNs, sometimes yielding Real Numbers.

A general Complex Number takes on the format:

The conjugated complex number of z, denoted as z<sup>*</sup> (“zed starred”) is:

So the conjugated complex number is obtained by inverting the sign of the complex part of the complex number (+ to – or – to +)

Via Euler, Complex Numbers are often expressed in their exponential form:

or

Trigonometry then establishes the relation between (r,θ) and (x,y): x = r cosθ, y = r sinθ

An important property of Complex Numbers (CNs) and their complex conjugates is that the product of a CN and its complex conjugate is a Real Number and positive, always:

(meaning z<sup>*</sup> times z) z<sup>*</sup>z = r e<sup>θi</sup> r e<sup>-θi</sup> = r<sup>2</sup> e<sup> θi – θi</sup> = r<sup>2</sup> e<sup>0</sup> = r<sup>2</sup>

In QM/WM this has an important application because the product of the wave function ψ (soon to be discussed, for all you hunky learners!) and its complex conjugate ψ<sup>*</sup> , that is ψ<sup>*</sup>ψ is always a Real Number.

Numerical Example: c = 5 - 3i, thus c<sup>*</sup> = 5 + 3i

c<sup>*</sup>c = (5 - 3i)(5 + 3i)

= 25 + 15i – 15i – 9i<sup>2</sup>

= 25 – 9 (-1)

= 25 + 9

= 34

Or for a general CN like c = x +/- yi,

c<sup>*</sup>c = x<sup>2</sup> + y<sup>2</sup>

[Edited on 10-7-2015 by blogfast25]

aga - 10-7-2015 at 08:14

That is the best and most concise explanation of i that I have ever heard or read.

Thanks bloggers.

blogfast25 - 10-7-2015 at 08:56

It's only the tip of a Complex Iceberg!

annaandherdad - 10-7-2015 at 09:06

Quote: Originally posted by Magpie |

A bit of history. In the 1870's Boltzmann applied classical thermodynamics, itself a relatively new development (Kelvin, Clausius et al, 1850's) to the problem of black body radiation. By doing a gedankenexperiment in which black body radiation was the working fluid in a Carnot cycle, he showed that the total energy in a given volume of black body radiation is proportional to T^4. The same had been observed experimentally by Stefan, and is now known as the Stefan-Boltzmann law; Stefan's experimental results gave a numerical value for the constant of proportionality. Boltzmann just used some simple thermodynamics, plus a bit of electromagnetic theory, the latter needed to show that the pressure is 1/3 of the energy density. The argument is simple and beautiful, and easily understood by a physics undergraduate today.

After this the main problem was to determine how the energy in black body radiation is distributed over frequency. Important advances were made by Wien, who by the early 1890's had derived his displacement law, which shows how the power spectrum of radiation scales with temperature, and his approximation to the power spectrum itself, which is accurate at high frequencies. Wien's arguments, unlike Boltzmann's, are somewhat complicated and not as interesting from a modern standpoint, but they represented real progress. In particular, Wien's high temperature formula shows the power spectrum (ie, the energy per unit frequency per unit volume in BB radiation) involves the quantity exp(-h*nu/k*T), where nu is the frequency, k is Boltzmann's constant and h is a new constant that we now call Planck's constant. This was the first appearance of Planck's constant in physics. Although Planck did not discover this constant, he did recognize that this was a new, fundamental constant of nature, on an equal footing with the speed of light, the charge on the electron, and Newton's constant of gravitation. Indeed, Planck realized that using this fundamental constant and the others one could construct a fundamental unit of length, time, energy, mass etc, units that are now called Planck units (the Planck length is about 10^{-33} cm, and it is the length scale at which the effects of both gravity and quantum mechanics are expected to appear). This was all before the correct distribution law was discovered.

A different theoretical argument, based on classical thermodynamics and a mode decomposition of the electromagnetic field, yields the Raleigh-Jeans law, which is valid at low frequencies, but which diverges at high frequencies. This divergence is called an "ultraviolet catastrophe".

Planck devoted himself to working out the formula for the power spectrum of BB radiation in the early 1890's, and worked on it for several years. His efforts focused on using electromagnetic theory to model the exchange of energy between matter and electromagnetic radiation. He worked with a kinetic equation describing the statistical flow of energy between matter and field, and attempted to find the equilibrium solution by a version of Boltzmann's "H-theorem", which was known to work for finding the equilibrium distribution of velocities in an ideal gas. Initially, Planck rejected Boltzmann 's statistical arguments, and had some sharp and public disagreements with him. Over time, however, Planck slowly began to realize that Boltzmann was right about statistics. This was a time in which the modern theory of equilibrium statistical mechanics was barely known (it had been developed recently by Gibbs, an American who wrote a book in English on his work. Gibb's work seems not to have been known to Planck). Later, in 1906, after Boltzmann had committed suicide, Planck expressed some regret and remorse, in a kind of acknowledgement of how much he owed to Boltzmann.

The breakthrough came in the fall of 1899, when Rubens and other experimentalists got some data on the low frequency end of the spectrum. Planck used this to interpolate between two formulas he had been playing with for the entropy of the BB radiation, and obtained the now-accepted formula for the power spectrum, which agreed with the experimental data.

Planck was lucky that he was in Berlin at the time, which is where the experimental work was going on. I wish I knew more about how Rubens and the others did it; I believe they developed effective bolometers, and I suspect that they used electric ovens to create the BB radiation.

The problem then was to justify the formula. In two months of furious work Planck did so, based on an ad hoc assumption that the exchange of energy between mechanical oscillators and the field could only take place in units of h*nu. The formula and the derivation were announced in December, 1899.

The story of how the wave-particle duality was discovered proceeds from this point. The main ideas were supplied by Einstein.

[Edited on 11-7-2015 by annaandherdad]

The Volatile Chemist - 10-7-2015 at 09:48

Great idea! Enjoying the explanation so far! May implement the key points into a program of some kind.

Polverone - 10-7-2015 at 12:48

Are you perhaps using McQuarrie and Simon's <i>Physical Chemistry: A Molecular Approach</i> as a rough outline? I ask because I have the pchem texts of Atkins, McQuarrie, and Levine but only the McQuarrie has a math chapter on complex numbers following the first quantum theory historical/background chapter. Or this could be simple coincidence.

The McQuarrie is my favorite of the 3 I have named above. It is the only one that builds up classical thermodynamics from quantum micro-foundations instead of following a more historical order of discovery.

blogfast25 - 10-7-2015 at 13:41

Quote: Originally posted by Polverone |

Unfortunately I can neither claim genius nor having read that textbook. It occurred to me to briefly introduce complex numbers under the assumption that most here would have only scant knowledge of them.

blogfast25 - 10-7-2015 at 13:42

Thanks AAHD!

neptunium - 10-7-2015 at 15:13

I remember a teacher long ago explaining to me complex numbers as a 3D volume, any point within it has an imaginary and a real coordinate . that helped me understand better...

blogfast25 - 10-7-2015 at 17:45

Quote: Originally posted by neptunium |

https://en.wikipedia.org/wiki/Complex_number

2D (the Complex Plane), yes. But 3D?

[Edited on 11-7-2015 by blogfast25]

aga - 10-7-2015 at 23:10

"Moreover, C has a nontrivial involutive automorphism x ↦ x* (namely the complex conjugation)"

*pop* there go some more brain cells.

i'll desperately cling onto this bit :-

"its complex conjugate ψ* , that is ψ*ψ is always a Real Number."

... and hope that when i hit an equation containing sqrt(-1) that it can be used to carry on to a real solution.

blogfast25 - 11-7-2015 at 04:31

Quote: Originally posted by aga |

ψ<sup>*</sup>ψ in other, related parts of math and QM is also known as the Inner Product.

blogfast25 - 11-7-2015 at 04:55

This is probably the most abstract part of the course so far and much will become clearer once we get to the Schrodinger equation. Students should nevertheless familiarise themselves with the following points. ‘Bookmark’ these prominently to make it easier to return to when necessary. For now the wave function will remain slightly mysterious.

At the end of this module another interlude to allow non-calculus people to catch a glimpse into Derivatives/Differential Equations.

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/wvfun.htm...

(Don't follow any links yet)

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/wvfun.htm...

(Don't follow any links yet)

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/qm.html#c...

(Don't follow any links yet)

QM/WM relies strongly on so called Differential Equations. In this course we’ll rarely (if at all) solve any of these but we’ll encounter some of them. So without going into Calculus, let’s have a look at what is a Derivative (First and Second) by means of a concrete example.

Take a simple function of x, like y = x<sup>3</sup>/10 + 2 = f(x) (“x raised to the power 3, divide that result by 10, then add 2”).

A few function values are x = 0; y = 2, x = 4; y = 2.8; x = -1, y = 1.9, obtained of course by plugging the x value into the function formula.

Below I’ve plotted y = x<sup>3</sup>/10 + 2 (blue) (as well as the First (amber) and Second Derivatives (grey)) for a range of x = - 5 to + 5). All three are smooth and continuous functions.

Note that the function f(x) is far from a straight line (the function is not linear), so that the slope (or gradient, syn.) of the curve changes as x varies. The slope can be visualised by taking a point on the curve at x and drawing a straight line through this point, parallel to the curve itself (a so called tangent). This can be repeated for each value of x.

But Calculus can be used to generate a general expression for the slope, called the First Derivative and which can be denoted as δy/δx (or δf(x)/δx)) or f’(x).

In the case of our example function:

y = x<sup>3</sup>/10 + 2

δy/δx = 3/10 x<sup>2</sup> = 0.3 x<sup>2</sup> = f’(x).

This new function, also denoted as f’(x) (“f prime of x”) or y’ can be plotted too, as shown on the graph in amber.

It needs to be understood that there is an f’(x) for each f(x). Math practitioners use tables with differentiating rules to tell them which is the First Derivative f’(x) of a given function f(x).

The new function f’(x) can itself be differentiated, resulting in the Second Derivative of f(x), namely f’’(x) (“f second of x”) or y’’, plot shown in grey.

In the case of the example f’’(x) = δ<sup>2</sup>y/δx<sup>2</sup> = 0.6 x

The Second Derivative is also noted as δ<sup>2</sup>y/δx<sup>2</sup> and this notation applied to the wave function ψ, as δ<sup>2</sup>ψ/δx<sup>2</sup>, will be frequently encountered in QM/WM.

Function ============ > y or f(x)

First Derivative ======= > y’; f’(x) or δy/δx

Second Derivative ===== > y’’, f’’(x) or δ<sup>2</sup>y/δx<sup>2</sup>

A Differential Equation is an equation in which First or Second (or Higher) Derivatives of a function occur.

[Edited on 11-7-2015 by blogfast25]

neptunium - 12-7-2015 at 03:22

go on Blogfast25! its been a while but its a good refresher! i like this..

lets get to the probabilty ..

Magpie - 12-7-2015 at 09:52

Quote: Originally posted by blogfast25 |

What's clear about pitchforks?

aga - 12-7-2015 at 12:00

Pitchforks are the only way to get a reluctant student to study properly IMHO.

Zombie's smoking behind the bicycle shed again, so we may as well get on ...

blogfast25 - 12-7-2015 at 13:36

Quote: Originally posted by aga |

You feel ready for it? Pitchforks coming up in spades, in that case!

[Edited on 12-7-2015 by blogfast25]

aga - 12-7-2015 at 13:50

Onwards and Sideways !

... as my one-legged Grandafther always said.

blogfast25 - 12-7-2015 at 14:10

I’ll be limiting our course to the time independent Schrodinger, which concerns itself with mainly Quantum Systems of constant energy. Also, for starters we’ll limit it to one dimension only.

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/scheq.htm...

Okay, so that’s not something you want to write on your Facebook wall. But what does it mean??

Let’s start with the right hand side of the equation:

Eψ(x)

This is the product of the particle’s Total Energy E and the wave function ψ(x).

Now look at the second term of the left hand side of the equation:

U(x)ψ(x)

This is the product of the Potential Energy U(x) and the wave function ψ(x). Note that here U(x) really is a function of x: over the range of x values where the particle ‘roams’ it cannot be assumed

Well, dear QMSMers, we’ve got a total energy related term on the right and one potential energy term on the left, rara, what could the left hand term be related to? Kinetic Energy perhaps?

Correct! The term:

- (ћ<sup>2</sup>/2m) δ<sup>2</sup>ψ(x)/δx<sup>2</sup>

(With ћ = h/(2π) (“h bar”))

… is related to the Kinetic Energy of the particle wave.

The Schrodinger equation is thus a kind of Energy Conservation Equation (E = U + E<sub>kin</sub> of Quantum Physics.

The time independent can easily be expanded into three dimensional domains (or two dimensional ones, for that matter).

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/sch3d.htm...

[Edited on 12-7-2015 by blogfast25]

aga - 12-7-2015 at 14:22

Woah ! This horse is going crazy !

This will take a bit of Time to digest, and even grab the ends of.

blogfast25 - 12-7-2015 at 14:29

Time will be granted. Try not to get lost in side issues (links) just yet.

Good night and good luck.

[Edited on 12-7-2015 by blogfast25]

aga - 14-7-2015 at 12:22

OK. Digested as much as i can, yet have to admit that i am a bit lost, already.

".. a kind of Energy Conservation Equation (E = U + Ekin)"

That makes a sort of sense, but where does it lead to ?

Some kind of roadmap of where we're going would be of great help.

blogfast25 - 14-7-2015 at 13:35

We're going many places but one very important one on the roadmap is to find out which states of total energy E are allowed for a given quantum system (atom, molecule or other). An equation (or equivalent of) like E = U + E<sub>kin</sub> is useful for that. It was Schrodinger's intention when he set up his famous equation.

The Schrodinger equation yields solutions for ψ, associated with allowable values of E.

ψ, in the case of an atom or a molecule gives us information on the whereabouts of the electron(s).

In the coming episode I'll use a specific quantum system to illustrate this.

Tell me when you're ready.

[Edited on 15-7-2015 by blogfast25]

Metacelsus - 14-7-2015 at 16:03

I have taken courses in linear algebra and vector calculus (but not Q.M.) so I know basically what the equation means mathematically. My question is: what does the constant factor (-Hbar^2/2m)

[Edited on 15-7-2015 by Cheddite Cheese]

blogfast25 - 14-7-2015 at 16:28

Quote: Originally posted by Cheddite Cheese |

It doesn't really matter, for my purposes here. You could rewrite the SE as:

... with

Again, it should become clearer as we go on.

Historical development of the SE:

https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation#Hist...

[Edited on 15-7-2015 by blogfast25]

blogfast25 - 15-7-2015 at 05:12

Although mainly a thought experiment (there exist some physical systems that resemble it, though) it’s an interesting one because despite its simplicity it shows properties shared by nearly all other (bound particles) Quantum Systems. We’ll use the system here as an example to show solutions of the Schrodinger equation and the properties of these solutions.

To make the system slightly more concrete, imagine the particle in a one dimensional box to be an electron, proton or other sub-atomic particle, trapped in a straight micro-capillary tube which has been sealed shut at both ends. The particle is wholly confined to the tube. In physical terms this means it would take an infinite amount of energy to ‘break out’ of this box.

Succinctly put, for a box from x = 0 to x = L ((“x between 0 and L”, box length L) the potential energy U(x) = 0 and for x < 0, U(x) = ∞ and for x > L, U(x) = ∞. The latter two conditions are also known as an ‘Infinite Potential Energy Well’. Movement of the particle is therefore restricted to the x- axis and between x = 0 and x = L.

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/pbox.html...

(Don’t follow any links on the above page yet)

Because of the U(x) = 0 for 0 <= x <= L (“x between 0 and L”) condition, the time independent one dimensional Schrodinger equation:

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/scheq.htm...

… is simplified enormously: simply eliminate the U(x)ψ(x) term because U(x) = 0.

The solution of this ‘reduced’ equation is A-level math but I won’t provide the derivation here, only the solution:

Ψ<sub>n</sub>(x) = √(2/L) sin(nπx/L) (for x = 0 to x = L)

Firstly, this isn’t just one wave function,

Ψ<sub>1</sub>(x) = √(2/L) sin(πx/L), Ψ<sub>2</sub>(x) = √(2/L) sin(2πx/L), etc.

Because each value of n (allowed values are 1, 2, 3, 4…, ∞, positive integers in other words! Zero is NOT allowed) defines a specific Ψ<sub>n</sub>(x).

I’ve plotted below the first 8 (n = 1 to 8) wave functions for the particle in a one dimensional box, schematically (y axis is not to scale).

The number n is known as the Quantum Number of the system.

1. The Schrodinger Equation (SE) for bound particles yields a set of wave functions Ψ<sub>n</sub>(x).

2. The SE yields a Quantum Number n, each allowable value associated with one wave function. n = 1, 2, 3, 4 etc.

3. The wave function Ψ<sub>0</sub>(x) = 0 is NOT allowed.

The last point merits mild elucidation. QM states that bound particles CANNOT be stationary. Even the ‘lowest’ allowed wave function (Ψ<sub>1</sub>(x) = √(2/L) sin(πx/L) ) is non-zero (except for x = 0 and x = L, of course)

Compare and contrast this with a Classical (non-Quantum) system: you can park your car and it will be stationary (E = 0) but that super small ‘quantum car’ would always be moving in its box! Quantum particles cannot ‘sit still’.

[Edited on 16-7-2015 by blogfast25]

aga - 18-7-2015 at 11:28

Sorry i'm late sir.

I was in detention all afternoon for holding Zombie's head down the toilet.

So by defining a restricted space, you can eleminate terms from the equation !

I like it.

More please : why cannot the Quantum bits remain still ?

blogfast25 - 18-7-2015 at 13:40

Quote: Originally posted by aga |

As long as it's not the old CaC2/toilet trick, I'll allow it.

U(x)Ψ(x) drops out because in this specific example U(x) = 0. But derivations for the particle in a box where U(x) is not zero do also exist. I chose U(x) = 0 because it's the simplest example.

Further down the course I'll show a Java applet that calculates wave functions for many other non-zero U(x) situations. All things in good time though.

A free electron can have v = 0 (E<sub>kin</sub> = 0) but that is a very different situation.

If you think all this is weird, you're right but wait till we get to quantum entanglement!

[Edited on 19-7-2015 by blogfast25]

blogfast25 - 18-7-2015 at 14:06

Many textbooks use an analogy that helps to ‘concretise’ (i.e. visualise, imagine) the wave functions somewhat. In it the matter waves of the particle are compared to the standing waves in a musical string (like a guitar string). You can generate and visualise standing waves in a real experiment by tying one end of skipping rope to a door knob (or such like) and moving the other end up and own with your hand. Get the frequency right and the skipping rope will take on the shape of Ψ<sub>1</sub>(x). Double the frequency of your hand movement and the pattern will resemble Ψ<sub>2</sub>(x) etc etc.

In this analogy the particle matter wave resonates like a standing wave in the box’ cavity, like the skipping rope forms a standing wave between the door knob and your hand.

But remember that moving particles

We’ve already seen that the probability P (P >= 0 and P <= 1) of finding that particle at location x is given by:

P(x) = ψ<sup>*</sup>(x)ψ(x)

With ψ<sup>*</sup>(x) the Complex Conjugate of ψ(x).

As it happens, we’re in luck! For the particle in a box all ψ<sub>n</sub>(x) are Real functions (ψ(x) returns Real Numbers for any real x).

And for a real number ψ, then ψ = ψ<sup>*</sup>. Because:

Take a real number ψ and write it as ψ + 0i (“zero times i”). Its complex conjugate is ψ<sup>*</sup> = ψ – 0i.

Obviously in this case ψ <sup>*</sup> = ψ and ψ <sup>*</sup>ψ = ψψ = ψ<sup>2</sup>.

Thus P(x) = ψ<sup>*</sup>(x)ψ(x) = ψ(x)<sup>2</sup> or simply ψ<sup>2</sup>, for Real wave functions.

So quite literally the probability P of finding the particle at location x is:

P(x) = ψ(x)<sup>2</sup>

We can now generalise this for any particle in a one dimensional box. Since as:

Ψ<sub>n</sub>(x) = √(2/L) sin(nπx/L)

It follows that:

Ψ<sub>n</sub>(x)<sup>2</sup> = (2/L) sin<sup>2</sup>(nπx/L) (or (2/L) [sin(nπx/L)]<sup>2</sup>, if you prefer).

I’ve plotted the Probability function P<sub>n</sub>(x) for n =1, 2 and 3 below. Note that the P(x) values are NOT to scale.

In WM and quantum Chemistry, the points where P = 0 are often referred to as ‘nodes’ and the areas where P > 0 as ‘lobes’.

[Edited on 18-7-2015 by blogfast25]

aga - 18-7-2015 at 15:23

Erm, can i digest this for a while ?

I have a feeling you'll extrapolate to three dimensions any second now, and it's quite hard for a dullard like me to keep a hold onto this maths train ride.

Thanks very much for explaining this, and for free.

Edit:

It feels like i should be paying for the time you're putting in to what looks like just My education.

[Edited on 18-7-2015 by aga]

blogfast25 - 18-7-2015 at 15:42

Quote: Originally posted by aga |

Take your time. We'll be staying in 1 dimension for a bit yet.

There will also be even less math in 3D because it's far too complicated for my purpose here.

aga - 18-7-2015 at 15:49

Thank you.

blogfast25 - 18-7-2015 at 17:46

Just let me know any questions you WILL have and when you're ready to move on.

Did you get Zomb's head out of the toilet pan or is he still smoking behind the gents?

[Edited on 19-7-2015 by blogfast25]

Darkstar - 18-7-2015 at 20:38

Quote: Originally posted by blargish |

I know this thread is geared more towards introductory-level physics, so this is probably going to be well beyond the scope of this discussion, but the reason macroscopic systems like trucks and bullets don't behave like quantum systems is not actually because of their size. All systems, regardless of size, are inherently quantum systems by nature and thus governed by quantum mechanics. The reason for the emergence of seemingly classical behavior from otherwise large quantum systems is due to the irreversible loss of coherence through interactions with their environment (decoherence). Being an open quantum system, the large number of degrees of freedom in the surrounding environment rapidly destroys any quantum properties of the object. This is why you don't see macroscopic objects in superposition.

By continuously being bombarded with photons and bumping into nearby air molecules and various other atoms/molecules, the truck is constantly being monitored or "observed" by its environment. The states of the system (the truck) quickly become entangled with the states of the environment, decohering them from one another and preventing interference. Through a process called einselection, "pointer states" are singled out. These states are the least prone to entanglement with the environment, and are the ones that remain after decoherence. These surviving states are essentially "classical states" that continue to persist despite the environmental monitoring, and are why macroscopic objects appear to have definite properties. Since it's usually the position basis that gets singled out, macroscopic objects tend to be in eigenstates of position. This transformation of a pure state into an improper mixed state gives the appearance of wavefunction collapse. This is why the truck doesn't exhibit "wavy" behavior.

The reason for classical behavior in macroscopic objects is not simply because of their size, but the difficulty in maintaining coherence in open quantum systems. A truck will never behave like a wave for this very reason. There are ways, however, such as using extremely cold temperatures, to slow the decoherence process down long enough to actually see quantum effects in macroscopic objects.

Quote: Originally posted by aga |

Don't worry, that's perfectly normal. Everyone's lost when they're first introduced to the strange and counter-intuitive world of quantum mechanics. To be honest, I'd be far more concerned if you

The irony of it all is that the longer you study QM, the less sense it actually makes.

aga - 19-7-2015 at 02:02

Quote: Originally posted by Darkstar |

Oh goody.

I'm way ahead of the game then !

blogfast25 - 19-7-2015 at 05:17

Thanks Darkstar, that was an interesting interlude.

One can also point to the 'Correspondence Principle' (I might spend a brief moment on it later on) to show that real (small) quantum systems start behaving 'Classically' for high values of E<sub>n</sub> (high n, in other words). The border between quantum and classical is fuzzy, not sharp.

Of QP has also been said that if you think you've understood it, that means you've not studied it enough! That might sound like QP is the lunatic asylum of physics but the 'problem' is that it works and provides real life solutions to real life physical problems! Doh!

[Edited on 19-7-2015 by blogfast25]

blogfast25 - 19-7-2015 at 10:00

Here’s that Java applet that shows energies (eigenvalues) and wave functions for a variety of ‘particle in a box’ situations:

http://www.falstad.com/qm1d/

Select “1-D Quantum Crystal Applet”.

In the new window the applet will start. Right top is a control panel that allows varying settings of the system. From the top drop down menu, select ‘Infinite well’.

In the top box the horizontal lines represent the energy levels (eigenvalues), the ground state is in red.

Mouse-hovering over the energy levels you’ll see the wave functions (in yellow) appear in the middle box.

Moving the slide rulers you can adjust L (box width) and m (particle mass) and observe the changes in E and ψ(x).

Now select ‘Infinite well + Field’ from the top draw down menu. This introduces a U(x) function that is non-zero and linear: U(x) =

Have fun.

[Edited on 19-7-2015 by blogfast25]

aga - 19-7-2015 at 12:21

Please Sir. Can't see an Infinite Well option Sir.

Got :-

Finite Well

Well Pairs

Couple Well Pairs

Harmonic

Coulomb-like

Free Particle

Sinusoidal

blogfast25 - 19-7-2015 at 12:55

No, I can see it: Set up: Infinite Well

It's the top option: move the right slide bar to the top.

You in IE or another browser?

However, you can look at Finite Well as well. That means the walls are not infinitely high but finitely high.

Screenshot:

[Edited on 19-7-2015 by blogfast25]

aga - 19-7-2015 at 14:02

That was Firefox.

Will try IE.

blogfast25 - 19-7-2015 at 14:47

Infinite well with Linear Field: wave function for n = 5:

White horizontal lines in top box are E<sub>n</sub> eigenvalues (red is ground state).

White sloped line is U(x) =

Yellow wavy line in second box is Ψ<sub>5</sub>(x).

[Edited on 19-7-2015 by blogfast25]

Darkstar - 20-7-2015 at 10:13

Quote: Originally posted by blogfast25 |

No problem. Just thought I'd contribute a little to the discussion since you're the only one participating at the moment. I think the idea of a free and interactive QM course here on SM is an awesome idea, and I'd be more than happy to help you teach. It's very generous of you to donate so much of your time tutoring others for free, especially when it's just one-on-one (like with aga). It's such a shame that so few here actually seem interested.

Quote: |

True. I actually considered mentioning the correspondence principle in my previous post since it helps to explain

So in your example of highly energetic quantum systems behaving classically, decoherence would offer an explanation as to why it happens. The high-energy system is constantly emitting entangled photons in every direction that end up interacting with the surrounding environment and causing the system to become entangled with it, effectively destroying the superposition (from our point of view, at least) and preventing any further interference. This also explains why extremely cold temperatures can allow for macroscopic systems to exhibit quantum behavior, as they aren't emitting entangled photons that would inevitably interact with the environment, allowing them to maintain coherence.

[Edited on 7-20-2015 by Darkstar]

blogfast25 - 20-7-2015 at 13:57

Darkstar:

Thank you. Also for your further elaboration.

'Enrolment' is lower than I hoped for but with now over 1,000 views the thread is probably followed a bit more closely than the number of commenters might suggest. But it was always going to be a bit of a thankless task.

[Edited on 20-7-2015 by blogfast25]

j_sum1 - 20-7-2015 at 15:19

Quote: Originally posted by blogfast25 |

Hey, I'm thankful. I haven't had much time to go over this closely but I have been reading and following some links. The java applet is not making its way through the aggressive firewall operational at work so I haven't had a decent play with the parameters. I think however you might be building a nice compact little tutorial here and one that I will pass on to y physics teacher colleagues.

Just because I m not vocal, don't assume I am not involved.

blogfast25 - 20-7-2015 at 15:25

Thanks, j_sum1, appreciated!

aga - 20-7-2015 at 15:27

It is necessary to Speak before the OP can know anything about your thoughts.

Thank you for doing so.

The notion that a member such as blogfast25 was investing the Time and Effort to teach just the Forum Drunkard was a bit difficult.

[Edited on 20-7-2015 by aga]

blogfast25 - 21-7-2015 at 07:15

Another quick tip re. the quantum applet :

http://www.falstad.com/qm1d/

Once you've selected the physical system (infinite well, for instance), from the second drop down menu select 'mouse = set eigenstate'. Also, slide the 'particle mass' slide rule somewhat more to the left (thereby reducing particle mass and the number of eigenvalues).

This way, when you hover over the top box and click on an eigenvalue (energy E<sub>n</sub> the probability function will also be shown, along with the wave function.

The applet also, somewhat inadvertently, demonstrates the Correspondence Principle briefly touched on higher up. If you drastically increase the particle's mass you'll notice that the energy levels become much closer together. In the limit (for higher and higher values) of m, the energy spectrum becomes an

[Edited on 21-7-2015 by blogfast25]

MrHomeScientist - 22-7-2015 at 05:43

I'm certainly following the thread. Don't be discouraged if there doesn't seem to be much interest up front - in a few months or years, someone might find this again and be inspired! Despite the lack of commenters, I'm sure many people are avidly reading.

I did have one question from back on page 2, when you said: "QM states that bound particles CANNOT be stationary. "

Could you elaborate on this more? Why exactly is psi = 0 not allowed?

Might it have something to do with the uncertainty principle? If so, I'd think that any particle (bound or unbound) could not be stationary since you'd know both its position and momentum.

Or perhaps it's because everything possesses wave-particle duality? So the particle is never really 'stationary' because that word doesn't have much meaning for a wave-like object.

This is probably something I should know, being a physics major and all :/

[Edited on 7-22-2015 by MrHomeScientist]

j_sum1 - 22-7-2015 at 06:38

Quote: Originally posted by MrHomeScientist |

I think these are alternate ways of saying much the same thing. (Please correct me if I am wrong.)

It reminds me of my professor in a course I did on solid state materials (semiconductors et al.) After speaking at length about the Heisenburg uncertainty principle, he suddenly asked a question on why we never find an electron in the nucleus. Silence from 200 or so electrical engineering students. He elaborated, "if an electron is negatively charged, it must be strongly attracted to the positive nucleus. You'd think that there would be a non-zero chance of finding it there." More silence. He then gave us the answer – "Because then we'd know where it is!"

Well, it was funny at the time. I don't suppose it types up very well.

It was then that the penny dropped for me. The uncertainty principle is not about our inability to find stuff and figure it out. It is about a genuine inability for objects to occupy a precise location or have zero velocity. It necessarily implies wave-like behaviour for quantum particles. I have had to retrain my brain in the way I think about electrons and their orbitals. I can't think of them as dots buzzing around randomly like spastic bees. Rather I have to think of them as being like waves in an aquarium of a particularly weird shape.

blogfast25 - 22-7-2015 at 06:39

Quote: Originally posted by MrHomeScientist |

For one, ψ = 0 also means ψ<sup>2</sup> = 0! Zero probability of finding the particle... And since as P = 0 also outside the box, this means effectively: NO PARTICLE, whatsoever!

This has also been confirmed empirically: even before QP we knew that the hydrogen atom has a lowest state of energy (the Ground State) that is not zero (see Bohr's Model and Rydberg).

In the hydrogen atom's ψ<sub>1</sub> corresponds to E<sub>1</sub>, the Ground State. ψ<sub>0</sub> would be where the electron has crashed into the nucleus (as predicted by Bohr's model). No wave function means no atom!

Thanks for reading!

Thanks also j_sum1, that is correct. There's more than one way of skinning a cat here!

[Edited on 22-7-2015 by blogfast25]

j_sum1 - 22-7-2015 at 06:58

I have the cat skin. I still don't know if the cat is alive or dead though.

On the topic of thread contributions, I think it is much better in this case to keep the signal to noise ratio high. And with that, I will shut up for a while.

blogfast25 - 22-7-2015 at 07:37

Quote: Originally posted by j_sum1 |

As long as you're not looking at her, she's both dead and alive!

[Edited on 22-7-2015 by blogfast25]

annaandherdad - 22-7-2015 at 10:22

Quote: Originally posted by Cheddite Cheese |

This constant factor is multiplied times the differential operator d^2/dx^2, that is, the term in the Schro"odinger equation is -(hbar^2/2m) x (d^2 psi/dx^2). This is regarded as (1/2m) times the square of the momentum operator p, where p is the differential operator -i hbar d/dx, that is, the term in the Schro"dinger equation is

-(hbar^2/2m) x (d^2 psi/dx^2) = (hbar/2m) x (-i hbar d/dx)^2 psi

= (p^2/2m) psi

(This is easier to understand if you write it out on the blackboard).

In classical mechanics, things you observe (energy, momentum, angular momentum, position of planet x) are just numbers, which may be functions of time in a particular system. In quantum mechanics, when you observe something (like the position or energy of an electron) you get a number, just like in classical mechanics, but unlike classical mechanics, the number you get isn't necessarily the same if you make the identical measurement on identically prepared systems. That is, there is a statistical distribution of the answers.

To calculate the statistical distribution of answers you need the linear operator associated with the observable. Every observable corresponds to a definite linear operator. The linear operator acts on wave functions psi and maps them into new wave functions. As I mentioned in the last paragraph, the linear operator corresponding to momentum is -i hbar d/dx.

How do we know this is the momentum operator? It was a guess of deBroglie. Einstein had been saying that a photon of energy E and momentum p was associated with a wave of frequency omega and wavenumber k, where E=hbar omega and p =hbar k. (omega and k are measured in radians; omega is radians/sec and k is radians/cm or radians per meter if you prefer). So deBroglie said, maybe this applies to massive particles as well. If so, a massive particle (like an electron) of energy E and momentum p must correspond to a wave of frequency omega=E/hbar and wave number k=p/hbar. That is, the wave must be

psi(x,t) = exp[i(kx -omega t)] = exp[i(px - Et)/hbar]

Now, deBroglie said, if we let the operator -i hbar d/dx act on this wave, we get

(-i hbar d/dx) psi(x,t) = p psi(x,t),

that is, the operator acting on the wave is the momentum times the wave. Historically this was how the momentum got associated with this particular differential operator. By the same logic, the energy is associated with the operatior (i hbar d/dt). (These are really partial derivatives, since psi depends on both x and t.)

To go back, the operator p^2/2m, which appears in one term in the Schro"dinger equation, corresponds to the classical number p^2/2m (where now p is a value, not an operator), which is otherwise mv^2/2 because p=mv, which is otherwise the kinetic energy. So, to answer your question, what this factor means physically, it means that this term in the Schro"dinger equation is the kinetic energy term. In fact, the Schro"dinger equation is an operator encoding of the classical energy relation,

p^2/2m + V(x) = E

blogfast25 - 22-7-2015 at 10:35

Thanks AAHD:

I was trying to avoid Quantum Operators because they're hardly simple, for those with minimum math knowledge.

I might dedicate a small segment to them at the end of Part 1.

aga - 22-7-2015 at 10:37

Quote: |

Phew. Head. Over. Whoosh !

blogfast25 - 22-7-2015 at 11:44

Quote: Originally posted by aga |

It looks a lot worse without proper scientific notation.

Gimme five.

aga - 22-7-2015 at 11:48

High Five Bro (?)

blogfast25 - 22-7-2015 at 11:54

psi(x,t) = exp[i(kx -omega t)] = exp[i(px - Et)/hbar]

Ψ(x,t) = e<sup>i(kx – ωt)</sup> = e<sup>i(px – Et)/ћ</sup>

It’s the Time Dependent Schrodinger equation for a free particle:

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/scheq.htm...

But in AAHD's formula the factor A (amplitude) is missing...

Neat, huh? [cough!]

[Edited on 22-7-2015 by blogfast25]

aga - 22-7-2015 at 12:00

Oh ! I see now !

[/lie mode off]

Hokay.

No. Just No.

I am very pleased that other people understand this very well, and can have fun and games with it.

Personally, i do not follow the maths at all, and i assumed the idea of this thread was to Teach rather than to be a discussion group for advanced theoretical mathematicians.

Yes, i am a drunken dullard because i do not grasp the idea even though i have been shown pitchforks, and should be left in the gutter where i belong.

If some Good Smaritan could summon some extra goodness, perhaps they could explain in simple terms what the Psi function is attempting to model, and how on earth you get from kx - ... to wx - ...

For starters, Schroedinger's equation for a time dependant particle in a matchbox is attempting to describe what exactly, how, and what does each term actually mean ?

[Edited on 22-7-2015 by aga]

[Edited on 22-7-2015 by aga]

blogfast25 - 22-7-2015 at 12:12

Or with Euler:

Ψ(x,t) = Ae<sup>i(kx – ωt)</sup> = A cos(kx – ωt) + Ai sin(kx – ωt)

Much better, huh?

blogfast25 - 22-7-2015 at 12:13

Quote: Originally posted by aga |

That's a tad unfair. I'm keeping it as math lite as possible. Hard to do with QP.

As I wrote above: I want to avoid the TDSE as much as possible. Forget about this interlude.

[Edited on 22-7-2015 by blogfast25]

aga - 22-7-2015 at 13:27

Quote: Originally posted by blogfast25 |

To re-iodiotate,

Please Explain.

Psi function of x and t =

unexplained A mutiplied by presumably Euler's number

to the power of the square root of -1 multiplied by some random k minus some small buttocks multipled by t.

Obviously the following error or failure to understand is laughable !

fcs = (uint32_t *) &buf[len];

*fcs = swaporder(crc32(len-RTAP_SIZE, &buf[RTAP_SIZE]));

Belly laughs all the way when the Fool did not realise he should do a simple cast of the buf pointer to a const * uchar_8 ! How could he NOT know !?!?! Gfaw Gfaw.

Oh the joy that the partial mastery of a single paradigm must bring.

Now, where were we with a simpler explantion of the actual terms being used in that equation, and how they were arrived at ?

Edit:

said pitchfork(x,y) instead of (x,t)

[Edited on 22-7-2015 by aga]

blogfast25 - 22-7-2015 at 14:07

OK, G-ddammit, that's it, detention for you:

Copy this text FIVE times, in LONGHAND:

https://en.wikipedia.org/wiki/Free_particle#Non-Relativistic...

Oral examination on the material tomorrow!

[Edited on 22-7-2015 by blogfast25]

annaandherdad - 22-7-2015 at 14:23

Quote: Originally posted by aga |

Sorry, aga, I was answering CC's question. I think different people here have different backgrounds. The formula, which bf25 made look much nicer, is the formula for a wave in complex notation. It's pretty standard in the theory of waves. deBroglie used it to connect the properties of a wave (its frequency and wave length) to the properties of the associated particle (its energy and momentum). I'll explain it if you want.

It was deBroglie's dissertation. Later Debye (a noted physicist at the time) commented that if there was a wave there had to be a wave equation. This motivated Schro"dinger to find his equation. It wasn't straightforward, because he tried to find the relativistic version of the wave equation first, and that's much more complicated. He gave up for a while, but later returned to the problem, this time looking for the non-relativistic version, which is the version usually known today as "the" Schro"dinger equation. The wave I wrote down is a solution for the case of a free particle.

aga - 22-7-2015 at 14:33

Sorry for the outburst.

One was simply trying to make clear how little can be understood when there's a lot of Running before Walking.

Explaining would be very helpful, but please try to use simple terms, such as

As a noob, i do not immediately recognise Qdangle as anything at all.

@Blogfast25

Morning Orals have been banned since 2002 following the Jenkins vs Blogfast24 affair if you may recall, Sir25.

blogfast25 - 22-7-2015 at 15:35

aga:

The physical meaning of

This should also help:

https://en.wikipedia.org/wiki/Matter_wave#de_Broglie_relatio...

And maybe this too:

https://en.wikipedia.org/wiki/Wave#Phase_velocity_and_group_...

In the example of a quantum system featured so far (particle in 1D box) time plays no part: the (time independent) SE explores the 'steady state' possible states of the system: E<sub>1</sub>, E<sub>2</sub>, E<sub>3</sub>, etc (and the associated wave functions Ψ<sub>n</sub>(x)) . We're not considering any CHANGES or how these occur. Change inevitably involves time (t) and the solving of the TDSE. Considering the added complexity of making the system

Hope you understand.

[Edited on 23-7-2015 by blogfast25]

aga - 22-7-2015 at 23:52

I think i was missing what the Result means.

Please stamp on any errors here :-

Ψ(x,t) computes a positive real number indicating the probability of a particle existing at position x at time t, given that it's Amplitude and frequency are known.

Ψ(x,t) = Ae<sup>i(kx – ωt)</sup> or much better

Ψ(x,t) = A cos(kx – ωt) + Ai sin(kx – ωt)

where :-

x = horizontal position

t = time

A = amplitude

ω = angular frequency (how fast the particle is spinning) given by 2πf where f = frequency in Hz

k = the 'wave number' = 2π/λ.

λ = c/f where c = the speed of light.

This was helpful :-

http://www.conspiracyoflight.com/Schrodinger/Schrodinger.htm...

blogfast25 - 23-7-2015 at 06:16

Ooopsie, not sure where this repetition came from.

[Edited on 23-7-2015 by blogfast25]

blogfast25 - 23-7-2015 at 06:17

aga:

One error: λ = c/f

That is true for electromagnetic waves, like light. But 'electron waves' are de Broglie waves. For matter waves (de Broglie waves) in general:

Now go back a bit to this:

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/pbox.html...

In the box, the matter waves are 'standing waves'. What are the wavelengths λ?

Well, you can literally see this from the schematic:

for n = 1, λ = L/2

for n = 2, λ = L

for n = 3, λ = 3L/2

for n = 4, λ = 2L

for n, λ = nL/2

That was a good link, aga. I can see from that link I will be able to skip the treatment of the hydrogen atom altogether!

[Edited on 23-7-2015 by blogfast25]

aga - 23-7-2015 at 08:10

Quote: Originally posted by blogfast25 |

Phew !

For absolute clarity :-

h being Plancks' constant right ?

and Ψ(x,t) does calculate a real number indicating the probability of finding the widget at x, t ?

Sorry if this is like wading through treacle for you, just that if i don't actually understand it, i just have to ask questions, or i never will understand it and be utterly lost.

blogfast25 - 23-7-2015 at 08:18

Quote: Originally posted by aga |

h is Planck's constant, indeedy.

The probability of finding a particle at x,t is given by: P(x,t) = Ψ(x,t)Ψ<sup>*</sup>(x,t)

With Ψ<sup>*</sup>(x,t) the Complex Conjugate of Ψ(x,t).

For a free moving particle Ψ(x,t) is a complex function but as we've seen above for any complex number a:

a<sup>*</sup>a is always a Real Number. P(x,t) = Ψ(x,t)Ψ<sup>*</sup>(x,t) always returns a Real Number. Probabilities are of course real numbers (0 <= P <= 1).

Applied to a free moving particle:

Ψ(x,t) = Ae<sup>i(kx – ωt)</sup> = A cos(kx – ωt) + Ai sin(kx – ωt)

Ψ(x,t) Ψ<sup>*</sup>(x,t) = A<sup>2</sup>

Another thing that might help you is this: simple sinusoidal waves:

https://en.wikipedia.org/wiki/Wave#Sinusoidal_waves

In essence a wave is a harmonic oscillation, travelling through space (x) at constant speed.

Take a tuning fork, for instance. Hit the fork and it starts oscillating at the pitch (frequency) of its note. The oscillating fork 'pushes' against the elastic air and a sound wave travels with that pitch, at the speed of sound.

[Edited on 23-7-2015 by blogfast25]

aga - 23-7-2015 at 09:18

Ah Right. I missed the i hiding next to the A before the sin, so Ψ(x,t) isn't a real number.

OK. Thanks. By jove i got it !

[Edited on 23-7-2015 by aga]

blogfast25 - 23-7-2015 at 09:26

You'll be relieved that the next instalment is lighter. [cough! Must take some Benyllin]

Darkstar - 23-7-2015 at 09:49

Quote: Originally posted by aga |

The fact that you're even trying to in the first place already puts you light years ahead of the general population. No one said these concepts were simple or easy to understand. Don't be so hard on yourself.

blogfast25 - 23-7-2015 at 09:58

Certificates of attendance will be handed out at the end of the course. Unless I lose the will to live before it!

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