Quantum Mechanics/Wave Mechanics in Chemistry Beginners Thread

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^st 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: Basic wave mechanics, up to atomic orbitals. I will use mainly HyperPhysics as an online textbook.

Part II: Applications of wave mechanics in Chemical Bond Theory, mainly from a VESPR perspective, but touching on most types of chemical bonds.

First instalment tomorrow.


[Edited on 9-7-2015 by blogfast25]

Quick Navigation (Updated):

Part I – Basic Wave Mechanics

• 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
  

Part II - Applications of Wave Mechanics in Chemical Bond Theory

• 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]

@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,

Quote: Originally posted by aga

So very high energy indeed. Hang the man.

Quote: Originally posted by aga

Why would a longer wavelength affect the accuracy ?

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 diffraction of the electron waves to occur.

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]

Quote: Originally posted by aga

I'll ignore all that then, and as a bonus, i'll not mention anything at all that undermines the entire paradigm until after the end-of-course exams.

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!

Quote: Originally posted by blogfast25

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

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!

A mini math-excursion: Complex Numbers

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 “i” frequently in the QM context, it’s worth to briefly elaborate. You might learn something, like a Xmas party trick or somefink.

At the heart of complex numbers lays the imaginary number i (“eye”):

i = √(-1) (the Square Root of minus one)

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 i² = - 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:

z = x + yi with x the Real part and yi the Imaginary (or complex) part, the choice of symbols x and y is entirely arbitrary and up to the author.

The conjugated complex number of z, denoted as z^* (“zed starred”) is:

z^* = x – yi

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:

z = r e^θi and its complex conjugate z^* = r e^-θi

or z = r cosθ + r sinθ i (or z = r cosθ + (r sinθ) i , to be perhaps clearer, as the sine acts on θ but not i)

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

Consequence for QM/WM:

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^* times z) z^*z = r e^θi r e^-θi = r² e^{θi – θi} = r² e⁰ = r²

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^* , that is ψ^*ψ is always a Real Number. Ψ(x)^*ψ(x) is related to the probability of finding the particle at location x and probability (P) is always a Real and Positive Number.

Numerical Example: c = 5 - 3i, thus c^* = 5 + 3i

c^*c = (5 - 3i)(5 + 3i)
= 25 + 15i – 15i – 9i²
= 25 – 9 (-1)
= 25 + 9
= 34

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

c^*c = x² + y²



And that’s it for this one, folks (you beautiful people ). Please, send us your questions, comments, corrections, magical incantations or moneys…

[Edited on 10-7-2015 by blogfast25]

Quote: Originally posted by Magpie

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?

Quote: Originally posted by Polverone

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.

Quote: Originally posted by neptunium

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...

Quote: Originally posted by aga

"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.

The Wave function ψ (psi, “pseye”) - The Heart of the Beast (at least for WM):

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.

The Wave function ψ

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

(Don't follow any links yet)

Wave function properties

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

(Don't follow any links yet)

Constraints on the wave function

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

(Don't follow any links yet)

Another brief math interlude: derivatives of mathematical functions

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 = x³/10 + 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 = x³/10 + 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.

Look at the slope a bit while imagining driving a car from x = - 5 to x = + 5. The first bit is the hardest: it's uphill and the slope is greatest. Things really easy off at around x = 0 which is a brief plateau. Then things get progressively harder again as slope increases continually from x = 0 to x = +5.

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 = x³/10 + 2

δy/δx = 3/10 x² = 0.3 x² = 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) = δ²y/δx² = 0.6 x

The Second Derivative is also noted as δ²y/δx² and this notation applied to the wave function ψ, as δ²ψ/δx², will be frequently encountered in QM/WM.

Summarising notation:

Function ============ > y or f(x)
First Derivative ======= > y’; f’(x) or δy/δx
Second Derivative ===== > y’’, f’’(x) or δ²y/δx²

Differential Equations

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]

Quote: Originally posted by blogfast25

The Wave function ψ (psi, “pseye”) - The Heart of the Beast (at least for WM): This is probably the most abstract part of the course so far and much will become clearer once we get to the Schrodinger equation.

Quote: Originally posted by aga

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

Time independent Schrodinger equation

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).

Note that the total energy is written as E and not E(x). E is not a function of x but a Real Number that can take on discrete values, known as Eigenvalues of the Schrodinger equation. Together with the right wave function ψ(x), the right value of E provide a mathematical solution of the Schrodinger equation.

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 a priori that U remains invariant of x. Later we’ll see some ‘real life’ Potential Energy functions that can apply to particles, including to an electron bound to a nucleus, in other words a hydrogen atom.

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:

- (ћ²/2m) δ²ψ(x)/δx²

(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_kin of Quantum Physics.

In three dimensions:

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]

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]

Quote: Originally posted by Cheddite Cheese

My question is: what does the constant factor (-Hbar^2/2m) physically mean?

Particle in a one dimensional box

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_n(x) = √(2/L) sin(nπx/L) (for x = 0 to x = L)

Pause. Stop the presses for a minute and let’s unpack this a bit.

Firstly, this isn’t just one wave function, it’s a whole bundle of them! An (theoretically) infinite bundle, to be precise:

Ψ₁(x) = √(2/L) sin(πx/L), Ψ₂(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_n(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. Hurray! We’ve found the first Quantum Number in our course. Others will follow.

From this thought experiment can already be concluded a few really important things:

1. The Schrodinger Equation (SE) for bound particles yields a set of wave functions Ψ_n(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 Ψ₀(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 (Ψ₁(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]

Quote: Originally posted by aga

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 ?

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.

Bound Quantum bits can't sit still because that is a consequence of the SE: Ψ(x) = 0 (for this particular problem!) is not an allowed solution to it. It's part of the "bizarrerie" that is QP. Take it or leave it!

A free electron can have v = 0 (E_kin = 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]

The Standing Waves Analogy and the probability of finding the particle at location x (Particle in a Box Continued)

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 Ψ₁(x). Double the frequency of your hand movement and the pattern will resemble Ψ₂(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 have a dual character: they are both waves and particles at once! Looking inside the box we WILL find the particle, even though it also behaves like a wave.

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^*(x)ψ(x)

With ψ^*(x) the Complex Conjugate of ψ(x).

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

And for a real number ψ, then ψ = ψ^*. Because:

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

Obviously in this case ψ ^* = ψ and ψ ^*ψ = ψψ = ψ².

Thus P(x) = ψ^*(x)ψ(x) = ψ(x)² or simply ψ², for Real wave functions.

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

P(x) = ψ(x)²

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

Ψ_n(x) = √(2/L) sin(nπx/L)

It follows that:

Ψ_n(x)² = (2/L) sin²(nπx/L) (or (2/L) [sin(nπx/L)]², if you prefer).

I’ve plotted the Probability function P_n(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]

Quote: Originally posted by aga

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]

Quote: Originally posted by blargish

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.

Quote: Originally posted by aga

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

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 weren't. As John Wheeler once said, "if you are not completely confused by quantum mechanics, you do not understand it."

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

Quote: Originally posted by Darkstar

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

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_n (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]

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]

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






White horizontal lines in top box are E_n eigenvalues (red is ground state).

White sloped line is U(x) = ux.

Yellow wavy line in second box is Ψ₅(x).

[Edited on 19-7-2015 by blogfast25]

Quote: Originally posted by blogfast25

Thanks Darkstar, that was an interesting interlude.

Quote: Originally posted by blogfast25

'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

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_n 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 energy continuum and that is a Classical property. So for very large masses, the particle starts behaving Classically again. The difference between 'quantum' and 'classical' is not a leap but a fuzzy line in the sand.

[Edited on 21-7-2015 by blogfast25]

Quote: Originally posted by MrHomeScientist

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.

Quote: Originally posted by MrHomeScientist

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 :/

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.

Quote: Originally posted by j_sum1

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

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

[Edited on 22-7-2015 by blogfast25]

Quote: Originally posted by Cheddite Cheese

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) physically mean? [Edited on 15-7-2015 by Cheddite Cheese]

Quote: Originally posted by aga

Quote: psi(x,t) = exp[i(kx -omega t)] = exp[i(px - Et)/hbar] Phew. Head. Over. Whoosh !

Or with Euler:

Ψ(x,t) = Ae^{i(kx – ωt)} = A cos(kx – ωt) + Ai sin(kx – ωt)

Much better, huh?

Quote: Originally posted by aga

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.

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. 1,2,3... click and you're back in the room!


[Edited on 22-7-2015 by blogfast25]

Quote: Originally posted by blogfast25

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

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]

Quote: Originally posted by aga

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.

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:

λ = h/p with p = mv

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!


I know you're finding all of this hard but when I had Erwin Schrodinger and Werner Heisenberg in my class all those years ago, they found it hard too. Not to mention the endless food fights between them!


[Edited on 23-7-2015 by blogfast25]

Quote: Originally posted by blogfast25

λ = h/p with p = mv

Quote: Originally posted by aga

Quote: Originally posted by blogfast25   λ = h/p with p = mv 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.

Quote: Originally posted by aga

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.

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