What Is the Role of the Hamiltonian in Quantum Mechanics?

[Xezlec]

and now you accuse the respondents of not understanding you and 'abusing' notation. FRom noobdom to expert ?

I apologize for offending you. I'm not very good at figuring out how to say things that doesn't make people upset. I've never had more than one or two friends in my life so all that social stuff is kind of a mystery to me.

If there is something about the notation that is physics-specific, then perhaps that is my problem exactly. I was assuming we were using standard mathematical notation. My mistake.

EDIT: Does it help any if I say that it wasn't you personally that I was accusing of abusing notation?

 

[Xezlec]

After accidentally pissing off people twice here in the span of about a week, and still not figuring out what I was trying to figure out, I think maybe this is a sign that it's time for me to take another break from social interaction for a while. I'll look at those books and come back in another few months if I'm still confused. I've taken enough of your collective time for now.

Thanks for the help anyway.

 

[Mentz114]

Xezlec,
your replies are gracious, please don't be upset. I guess I rather over-reacted.

Good luck with the studies.

M

 

[malawi_glenn]

Xezlec:

"1. What is the formula for the probability distribution of a quantity that is momentum times position, given an arbitrary wavefunction \Psi"


"2. What is the formula for the probability distribution of a quantity defined by e^{p \over p_0} + {x \over x_0}?"

Why do you want to know all of this before you haven't even learn the basics yet?

You write you wavefunctions and operators in different 'representations', so you don't mix things like p and x, this will become clear if you study the first chapter of Sakurais book 'modern QM'. Also you work with observables in QM, if you want to construct states that are eigenstates of the operator x-i\hbar\frac{\partial}{\partial x}, but then you'll see that this will depend on WHAT order, you measure this, since the commutator [x,p_x] is not equal to zero...

So I would say that will not ask those kinds of questions if you study a bit QM first.

 

[reilly]

Xezlec: Hamiltonians are defined in classical mechanics. Basic Stat Mech and Kinetic Theory will be very useful in studying QM, particularly as an example and exercise in abstract thinking. Much about waves and partial differential equations can be learned by studying E&M. Further, basic mathematical physics deals with PDE's other than those found in E&M, deals with Fourier integrals, and so forth. You really need to master these subjects before you are ready for QM - otherwise the math will defeat you. The issues raised here are usually covered fairly early in most QM courses, as in most QM texts. Best to do some homework.
Regards,
Reilly Atkinson

 

[adartsesirhc]

Xezlec, don't despair! Quantum mechanics is a difficult subject to learn for the first time, since it's such a dramatic separation from everything we know so far. Maybe you should try to prepare for QM by learning some more math and physics. I'm learning QM on my own right now, and I eventually want to take Stanford's Education Program for Gifted Youth (EPGY)'s Intro to QM class. Here's the preparation I've had:

Math
1. High school calculus
2. Multivariable and vector calculus
3. Linear algebra
4. A little bit of the calculus of variations
5. A little bit of differential equations - enough to solve first-order and some second-order equations

Physics
1. Mechanics
2. Electricity and magnetism
3. A little bit of intermediate classical mechanics - just Langrangian and Hamiltonian mechanics

I'm using Griffith's Introduction to QM, and I've been doing pretty good. However, there are some bumps in the road occasionally, so I would recommend Stanford's pathway to QM:

Math
1. Multivariable and vector calculus
2. Linear algebra
3. Differential Equations
4. Real analysis
5. Complex analysis
6. Partial differential equations

Physics
1. Mechanics
2. Electricity and magnetism
3. Light and heat
4. Modern physics
5. Intermediate classical mechanics (including the calculus of variations and tensor analysis)

This is probably a much more thorough and complete pathway to QM - all of these courses are required to take EPGY's Introduction to Quantum Mechanics, which is "a rigorous introduction to the theory of quantum mechanics."


Textbook-wise, I would recommend Griffiths as the absolute best for a beginner in QM (and EPGY uses it as the course text), but it does have one drawback - in the introduction, Griffiths talks about the use of mathematics and operators in his book - "My own instinct is to hand the students shovels and tell them to start digging. They may develop blisters at first, but I still think this is the most efficient and exciting way to learn." While this is great for students who want to avoid boring and tedious lectures about the proper use of math and operators, it might not be best for someone who wants to understand the math very well, as you do. Hence, my conclusion: Try Griffiths. If it isn't for you, read Shankar or Sakurai, but be warned - the latter two are usually reserved for graduate classes.


As for your questions, replace every p with the momentum operator, \frac{\hbar}{i}\frac{\partial}{\partial x}, and every x with the position operator, X. I don't know what p_{0} and x_{0} are. Then, if you have a function with the operators, use a Taylor series expansion instead. =]