A few quantum mechanics problems

[AhmedEzz]

I just finished my Quantum Mechanics module and i have somethings that i already discussed with my professor but i didn't get what he was saying.

Does the quantum model of the simple pendulum "exist" as in real or verified experimentally?

because in the quantum model, there are some places where the probability of finding the pendulum at some places is zero..it simply doesn't make any sense to me.

one more thing , the quantum model of the SP suggested that the pendulum crosses the classical border/boundary...has this been verified experimentally ? if yes then under what conditions does this happen?

Last but not least, what solution to the time-independent Schrodinger Equation leads to the above?

I have been in the search function but after 20 mins i couldn't find what i was asking about. BTW, I'm in first year electrical engineering.

 

[blechman]

Does the quantum model of the simple pendulum "exist" as in real or verified experimentally?

there are 2 answers to this question:
1) SHO is one of the few problems in QM that has an exact, closed form solution, and therefore is a great place to teach students the basics.

2) For (appropriately well-behaved) potentials we can Taylor expand. Let's imagine that we are expanding around an equilibrium point, so the first derivative of the potential vanishes and the second derivative is positive. Then as long as you are able to treat the higher-order terms as "small", then you are looking at a SHO! For this reason, this model appears in almost every branch of physics! This is a very powerful "starting point" for trying to model real-world physics.

because in the quantum model, there are some places where the probability of finding the pendulum at some places is zero..it simply doesn't make any sense to me.

yeah, QM is nuts, isn't it? Get over it!

Seriously - there should be nothing wrong with the idea that for certain energy states, the particle might not be in some places. Also, you have to be careful about interpretation: when the wavefunction vanishes, it does **NOT** mean that the particle can never be there - what it means is that you have ZERO CHANCE of SEEING the particle at that point. That's by no means the same thing!

one more thing , the quantum model of the SP suggested that the pendulum crosses the classical border/boundary...has this been verified experimentally ? if yes then under what conditions does this happen?

oh, sure! this is VERY common in quantum mechanics. The idea of partlcles getting past the classical boundaries (called "tunneling") is a vital mechanism behind many phenomena. Since you're an EE person, I'll give you one of an infinite number of examples: SEMICONDUCTORS! That's how they work. Well...ok, they're complicated, but tunneling is part of it. I'll give you one more: alpha decay of a nucleus. I can go on...

Last but not least, what solution to the time-independent Schrodinger Equation leads to the above?

what do you mean by this?! the TISE is the equation that the wavefunction satisfies, and all these effects follow from that wavefunction! I don't get this question.

 

[AhmedEzz]

Well, i know that there are different solutions to the TISE for different systems, i.e. particle in free space or a particle in a box...so what I'm asking is that what is the solution of TISE that lead to these conclusions/tunneling ?

 

[AhmedEzz]

what it means is that you have ZERO CHANCE of SEEING the particle at that point. That's by no means the same thing!

again is this true? i mean in real life, i can't simply see the simple pendulum in some places, i know it sounds naive but please bare with me...by seeing i mean visually seeing it.

this is VERY common in quantum mechanics. The idea of partlcles getting past the classical boundaries

again ideas, what I'm talking about is real life, because if what you're saying is true than how did the world accept the idea of the classical boundary in the first place - given that it was tested in the lab-..?

sorry for double post.

 

[blechman]

Well, i know that there are different solutions to the TISE for different systems, i.e. particle in free space or a particle in a box...so what I'm asking is that what is the solution of TISE that lead to these conclusions/tunneling ?

any time you have a "classical barrier" the wavefunction can go through it, as long as it's not an infintite barrier (particle in a box). so if it's a finite square well, for example. Or the simple harmonic oscillator, as you pointed out.

again is this true? i mean in real life, i can't simply see the simple pendulum in some places, i know it sounds naive but please bare with me...by seeing i mean visually seeing it.

you have to be a little careful about verbs like "see" when talking about QM. For a classical SHO, it follows a well-defined path that you can trace. but a quantum SHO does not do this. It bounces around pseudo-randomly, its location being a random variable with a probability distribution given by the wavefunction. It's a *very* different kind of system. It takes some getting used to.

again ideas, what I'm talking about is real life, because if what you're saying is true than how did the world accept the idea of the classical boundary in the first place - given that it was tested in the lab-..?

sorry for double post.

well, semiconductors were discovered in the 20th century, as was alpha decay, and all the other examples I can think of. when these things were discovered, we either already had QM, or better yet: we were realizing that we needed QM! But remember that the tunneling length is typically VERY small - the wavefunction is an exponential decay. So until we had the sophisticated equipment to see such small things, they just didn't happen.

For example: for the CLASSICAL SHO to suddenly jump to twice it's natural height (the classical version of tunneling) - the quantum probability for this to happen is something like $1/10^{10^{100}}$ - and so it just never occurs.

I know sometimes I can get a little carried away. I am not trying to be discouraging to you. Keep asking your questions. They're actually very good questions for a beginning student!

 

[blechman]

about the particle never being found in certain places: that is actually much easier to comprehend when you think of the wave-picture. Remember the double-slit experiment: there are bands of darkness where the particles never hit the screen. Thought of as a wave, this follows from the usual interference. Thought of as a particle, it boggles the mind!

Even though this is not the same as Yong's double-slit experiment, the concept is identical - thought of as a wave, the particle in the SHO undergoes destructive interference at certain points.

This is why Feynman used to say that the double-slit experiment is the single greatest experiment ever - it has all the concepts of QM built into it!

 

[AhmedEzz]

well, thanks that's a relief but i want to be sure that i get what you're saying...
So the simple pendulum experiment that is in the lab actually works by the laws of classical physics and there's no need for QM there?

If that's true then why do you think i took the simple pendulum experiment in the lab? i know it has something to do with Simple harmonic motion and small oscillations..but I'm not sure

Thank you for your patience, i really appreciate this because if someone like you didn't help than too bad because i don't think i would have got the answer...as i said my professor's discussion is a bit too advanced for me atleast.

 

[blechman]

well, thanks that's a relief but i want to be sure that i get what you're saying...
So the simple pendulum experiment that is in the lab actually works by the laws of classical physics and there's no need for QM there?

correct. perhaps more technically correct: the QM effects all cancel each other out and the final result is Newton's laws of motion.

If that's true then why do you think i took the simple pendulum experiment in the lab? i know it has something to do with Simple harmonic motion and small oscillations..but I'm not sure

What do you mean? The simple pendulum is an example of a simple harmonic oscillator - that is to say, the potential energy is quadratic in the displacement. Another example is a spring (Hooke's Law). These are real, physical systems. Of course, they are only APPROXIMATELY simple harmonic oscillators: both systems have higher powers of the displacement (high-swinging pendulum, non-ideal spring), but we are allowed to set these to zero as an approximation.

I do not know of an EXACT quantum simple harmonic oscillator, but again, there are many systems that behave this way to a good approximation. One example is a diatomic molecule - to a first approximation, you can imagine that the two atoms are solid spheres connected together by a spring. This is VERY crude, but believe it or not, it does very well for a first approximation. Using more advanced methods ("perturbation theory") we can build on this approximation to do better and better. But this is more advanced.

Hope that helps!

 

[Ben Niehoff]

About the Quantum Harmonic Oscillator, to get solutions where the pendulum "travels" from one end to the other, you need to superpose a few energy eigenstates. Here's a neat little applet that explains it nicely:

http://web.ift.uib.no/AMOS/MOV/HO/

As you see, the particle in the QHO can indeed travel back and forth in sinusoidal motion, and occupy every location between the boundaries of the potential well.

 

[AhmedEzz]

Thanks a lot for blechman and Ben niehoff, you really helped me alot...i was under the wrong impression that i should compare classical and quantum model of the simple pendulum hence the weird questions but many thanks to you guys.

 

[blechman]

I feel a little silly not thinking of coherent states - good thinking, Ben! These are the classical oscillators in the quantum world - the state's "quantum numbers" are the expectation values of the position and momentum of the oscillator, so that's as good as you can get. But the explicit energy states do not have a simple classical analog.

Glad to hear this was helpful.

 

[jtbell]

As you see, the particle in the QHO can indeed travel back and forth in sinusoidal motion, and occupy every location between the boundaries of the potential well.

I would rather say that the particle's probability distribution travels back and forth in sinusoidal motion. Whether the particle itself travels back and forth in sinusoidal motion is a different question, which QM doesn't give an answer for.

 

[Ben Niehoff]

Sorry, I was conflating terms, using the word "particle" to mean "localized wave packet".