How Do Eigenvalues Relate to Quantum Mechanics and Classical Physics?

[CSOleson]

Hey all, I am learning about Quantum Mechanics right now and I have a few questions:

1. First off, can someone verify eigen functions and values for me. Why are they important and what exactl do they do?
2. How about is it possible when accounting for the 3rd postulate of quantum mechanics, dealing with the said eigen values, apart from the descrete energy values is there any possibility of a result of energy identical or agreeing with classical mechanics?
3. Why do we need a single value for these wave functions? Is this because without it particles would have a dual probability along a same given interval?

Thank you in advance for your help. I am taking a 4 credit hour course in physical chemistry on quantum mechanics where the entire course is being taught in a two month period, so it is like a 8 credit hour course. I have my first test in a few days and I am struggling with the information right now.

 

[atyy]

1. For some bizarre reason, something you can measure, like position, is associated with an operator. Take that on faith.

An operator does something to a function, usually something complicated. But if an operator corresponds to an observable, there are some functions on which the operator's action is just multiplication by a number. These function are the eigenfunctions of the operator, and the number by which that eigenfunction is multiplied is the eigenvalue of the eigenfunction.

The eigenvalues are important because these are the possible values one actually gets when one makes a measurement of an observable. When the measurement is made, if a particular eigenvalue is observed, the wave function immediately after the measurement is the eigenfunction corresponding to that eigenvalue.

2. One correspondence with classical mechanics is via Ehrenfests theorem, which says the average value of an observable has the same dynamics as it would have classically.

3. The wave function is the state of the system, and a system cannot be in two states at the same time.

 

[bhobba]

Why do we need a single value for these wave functions? Is this because without it particles would have a dual probability along a same given interval?

See the following for the conceptual core of QM:
http://www.scottaaronson.com/democritus/lec9.html

Basically its a certain generalization of probability theory and the state has a similar role to probabilities which have single values.

Thanks
Bill

 

[CSOleson]

Hey guys, another question:
I don't get quantum numbers. No, I am not talking about n, l, s, ml, and ms. I am talking about
L, S, ML, MS, and J. These are related to quantum states for many electron atoms and electron spectroscopy. I have a quiz over that tomorrow (term symbols) as well as transition and allowed spectroscopy. Do you think you can help me out? Thanks!

 

[dipole]

1. For some bizarre reason, something you can measure, like position, is associated with an operator. Take that on faith.

That's not really bizarre... it's an axiom of the theory, and so observables are represented by (Hermitian) operators by construction. No faith involved, it is a choice to build the theory in that way and one which happens to work beautifully.

 

[bhobba]

That's not really bizarre... it's an axiom of the theory, and so observables are represented by (Hermitian) operators by construction. No faith involved, it is a choice to build the theory in that way and one which happens to work beautifully.

Thats correct.

Check out:
http://www.scottaaronson.com/democritus/lec9.html

Basically QM is an extension of probability theory where the 'probabilities' are not positive numbers, but actually complex. The reason is if you want continuous transformations that's what you need to do - see the section on real vs complex numbers.

Now take one of these vectors and let |bi> be it's basis. It turns out for so called mixed states its better to consider operators |bi><bi| rather than vectors - this is because if you multiply the vector by a phase factor complex number, it makes no difference. Each of these operators are possible outcomes. Its simply a hop skip and a jump to get observables from this. If yi are the possible outcomes then we can define an operator O = ∑ yi |bi><bi|. This is a Hermition operator whose eigenvalues are the possible outcomes and eigenvetors are the |bi><bi|.

It's simply a neat and nifty way of encoding this.

Thanks
Bill

 

[CSOleson]

The following is an idea that I need to solidify for an upcoming test:

How do I properly define Centrifugal Potential?