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Prerequisites of Quantum Mechanics

by asen7
Tags: mechanics, prerequisites, quantum
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asen7
#1
Apr2-11, 02:39 PM
P: 13
Hi everyone,

I am a college freshman, and was interested in this topic. So I was wondering, where does one draw the distinction between the physics and chemistry of quantum mechanics? Or in other words, what topics do quantum chemistry encompass, and what does quantum mechanics/physics encompass?

I was also wondering, what are the science class and math class prerequisites for quantum mechanics?
And I am a bit new to college, so could you list the order of math one takes to cover all the math prerequisites (from the introductory classes to the more advanced classes)?
And the order of the science classes to cover the science prerequisites?

Thank you
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protazoa
#2
Apr2-11, 02:48 PM
P: 2
Quantum chemistry mathematically describes the fundamental behavior of matter at the *molecular* scale

Quantum physics is a branch of physics providing a mathematical description of the dual particle-like and wave-like behavior and interaction of matter and energy at the *atomic or subatomic* scale

as for the classes required, I cannot help you there: Im not in college yet
asen7
#3
Apr2-11, 03:10 PM
P: 13
Thanks,

Anyone know the classes/prerequisites though?

The_Duck
#4
Apr2-11, 03:26 PM
P: 865
Prerequisites of Quantum Mechanics

On the physics side, you should first take your school's freshman physics sequence. On the math side, you need multivariable calculus. Differential equations would be very helpful but you can probably learn what you need as you go. Helpful but probably not strictly necessary are linear algebra and Fourier analysis.
asen7
#5
Apr2-11, 03:36 PM
P: 13
is fourier analysis a seperate math course?
oraclelive
#6
Apr2-11, 05:50 PM
P: 25
On the physics, you'll need to have foundation knowledge of how a wave/particle behaves or should behave, wave equations of different wave functions. Fourier series may/may not be a different math course depending on your school curriculum.
Penn.6-5000
#7
Apr2-11, 06:30 PM
P: 31
Your college's registrar should tell you which classes are prerequisites for quantum mechanics classes. If you want to know what subjects you have to understand in order to follow quantum mechanics, here's a decent list:
1. Be comfortable manipulating complex numbers.
2. Understand linear algebra. Understand what it means to say "orthogonal matrices represent rotations" and "symmetric matrices represent inner products." Know what eigenvalues are. When you mix complex numbers and linear algebra, you get "unitary matrices represent rotations" and "hermitian matrices represent inner products."
3. Understand more linear algebra. What's a basis? An orthonormal basis? A change of basis? What does it *mean* when we say that a unitary matrix changes one orthonormal basis into another?
4. Eventually, understand even more linear algebra---what's the matrix exponential? What's the spectral theorem? What do they *mean*?

When I was first trying to figure out how quantum mechanics worked, I read about how Schrodinger's equation was solved for the hydrogen atom and figured that differential equations were really important. It turns out that they aren't; the harmonic oscillator, the particle in a box, and the hydrogen atom are really the only three differential equations that anybody ever seems interested in solving, and the solutions are generally handed to you from the start. Once the solutions are handed to you, then you apply a bunch of linear algebra techniques to combine and interpret them. So linear algebra is where most of the real insight lies.
asen7
#8
Apr2-11, 06:52 PM
P: 13
Quote Quote by Penn.6-5000 View Post
When I was first trying to figure out how quantum mechanics worked, I read about how Schrodinger's equation was solved for the hydrogen atom and figured that differential equations were really important. It turns out that they aren't; the harmonic oscillator, the particle in a box, and the hydrogen atom are really the only three differential equations that anybody ever seems interested in solving, and the solutions are generally handed to you from the start. Once the solutions are handed to you, then you apply a bunch of linear algebra techniques to combine and interpret them. So linear algebra is where most of the real insight lies.

So will I need much multivariable calculus? or vectors?

And the hamiltonian operators and eigenvalues, and other operators can be found in which level of math textbook? Or will it be found in a quantum mechanics textbook?
romsofia
#9
Apr2-11, 07:13 PM
P: 271
Quote Quote by Penn.6-5000 View Post
Your college's registrar should tell you which classes are prerequisites for quantum mechanics classes. If you want to know what subjects you have to understand in order to follow quantum mechanics, here's a decent list:
1. Be comfortable manipulating complex numbers.
2. Understand linear algebra. Understand what it means to say "orthogonal matrices represent rotations" and "symmetric matrices represent inner products." Know what eigenvalues are. When you mix complex numbers and linear algebra, you get "unitary matrices represent rotations" and "hermitian matrices represent inner products."
3. Understand more linear algebra. What's a basis? An orthonormal basis? A change of basis? What does it *mean* when we say that a unitary matrix changes one orthonormal basis into another?
4. Eventually, understand even more linear algebra---what's the matrix exponential? What's the spectral theorem? What do they *mean*?

When I was first trying to figure out how quantum mechanics worked, I read about how Schrodinger's equation was solved for the hydrogen atom and figured that differential equations were really important. It turns out that they aren't; the harmonic oscillator, the particle in a box, and the hydrogen atom are really the only three differential equations that anybody ever seems interested in solving, and the solutions are generally handed to you from the start. Once the solutions are handed to you, then you apply a bunch of linear algebra techniques to combine and interpret them. So linear algebra is where most of the real insight lies.
This. The language of quantum mechanics is really linear algebra!
twofish-quant
#10
Apr2-11, 09:44 PM
P: 6,863
Quote Quote by romsofia View Post
This. The language of quantum mechanics is really linear algebra!
It's also multi-variable calculus!!!

One cool thing about QM is that you find out that multi-variable calculus and linear algebra can be two different ways of looking at the same thing.
twofish-quant
#11
Apr2-11, 09:46 PM
P: 6,863
Quote Quote by asen7 View Post
So will I need much multivariable calculus? or vectors?
Not really. You'll need to be comfortable talking in that language. You won't need to do proofs.

And the hamiltonian operators and eigenvalues, and other operators can be found in which level of math textbook? Or will it be found in a quantum mechanics textbook?
Depends on the specific class.
asen7
#12
Apr3-11, 01:32 AM
P: 13
This is more of a self-study for me.

So can anyone recommend any good (as in very specific, detailed, clear, and understandable instruction) quantum mechanics textbooks preferably from introductory quantum mechanics textbooks to a few "classes" a bit more advanced than that?

Thanks everyone, I really do appreciate the responses so far.
hadsed
#13
Apr3-11, 01:35 PM
P: 492
Griffith's is normally a really good place to start. I read the Feynman lectures for a different perspective on things, and they're meant to be somewhat introductory as well.
hitmeoff
#14
Apr3-11, 02:08 PM
P: 261
Not sure why Griffiths is always recommended as a first read text on QM. Griffiths is a full blown junior/senior level text.

QM is quite different from classical mechanics and classical E&M, so in theory you don't NEED to go through those courses before tackling QM but you probably wont get much of anything from a text like Griffiths without first developing some sort "physics" maturity.

Much like abstract algebra truly doesn't require much more than basic Elementary algebra to understand, in reality you need some sort of exposure to rigorous proof techniques to succeed in abstract algebra; in terms of physics, this mean you should have some training in how to solve physical problems beyond the "plug and chug" variety you are likely to encounter in very basic freshman physics.

Your first stop should be your current freshman level text which probably (but may not) has a section on "Modern Physics" which will include a basic intro to QM without using much higher level math. From here you can move on to texts like Morrison's Understanding Quantum Physics or Modern Physics by Taylor (more formal textbookish), or for an idea of how matrices and such are used in QM you can try the (very cheap) "Quantum Mechanics in Simple Matrix Form".

On the math side, you will probably want to brush up on basic, elementary linear algebra. You don't need a math major's perspective on the subject so you don't need the rigor of a book like Axler, Friedberg, etc...Anton's Elementary Linear Algebra should suffice. As afar as ODEs/PDEs are concerned...not sure what I can recommend here. Coddington's Introduction to ODEs is very clear, straightforward and best of all cheap. I personally don't know of any "basic" PDE book (the only ones I know are rigorous and meant for math majors), but a good "overview" of the subject (as well as most of the basic math you will need for most junior/senior level physics classes) can be found in Boas.

This should give you the basic tools you'll need to tackle Griffiths. But remember most schools like to see you take Classical Mechanics and E&M at the junior/senior level before you take QM. The reason is not because you are going to carry over Newton's Laws or Coulomb's Law, but rather because these are THEORY based courses (vs. plug and chugg/application based physics in freshman physics). These courses force you to do problems for which you really have to understand the concepts and get creative with your problem solving skills in order to solve those problems. The benefit here is that classical E&M and (especially) Classical Mechanics are very physically intuitive which lets you fall back on your intuition to help you solve problems.

When you get to QM at the Griffiths level, things aren't as intuitive, but at least if you've been through a theory based course and can handle going from Theory to Problem-Solving more easily. If you are interested in seeing some theory based books Taylor's Classical Mechanics is the easiest of the bunch to read on your own. The standard junior level book on E&M is Griffiths (but be forwarned, this like his QM book is a full blown junior/senior level E&M book and does require you to really have you vector-calculus and PDE skills tight.), this can book also gives you a nice glimpse at the level the Griffiths QM book is written at, and the type of problems you can expect.

So long story, still long: your current freshman physics book, Morrison's book and QM in simple matrix form are your best bet to getting a nice feel for the subject without needing any extra training beyond what you have. If you want to go any deeper than these texts (i.e. Griffiths) you will probably want to do a bit more work (both in math and physics).
Penn.6-5000
#15
Apr3-11, 02:46 PM
P: 31
Volume three of the Feynman Lectures does an excellent job of explaining the intuitive underpinnings of quantum mechanics---what superpositions are, what the hamiltonian does, what changes of basis are, and so on. The downside is that Feynman takes his time in getting to how his explanations relate to the explanations you'll see in other books. For example, he introduces the concept of "stationary states" (which everyone else calls "eigenstates") in section 7-1. He first uses stationary states indirectly to solve a problem in section 8-6. In section 9-1 he explicitly computes the stationary states of the ammonia molecule. But it isn't until section 11-6 that he finally says, "Physicists usually call the states |n>
'the eigenstates of H.' " So he's strung you along for four chapters explaining what eigenvectors are and how to use them, but unless you're particularly clever you don't realize that you're using eigenvectors until section 11-6. So it can take awhile to relate Feynman's explanations to all the other textbooks, even though they eventually link up.

The upside to Feynman is the same thing---that he spends four chapters explaining how eigenvectors work before dropping the E-word. A lot of other books just assume that you're familiar enough with linear algebra to know what eigenvectors are, so they'll just use eigenvector methods without telling you why. Feynman assumes nothing---he teaches you how to multiply two matrices in section 11-1. So in my opinion Feynman is by far the best place to get your linear algebra intuition.

I wish I could recommend a good linear algebra book but I'm not particularly enthusiastic about any of them. I first learned linear algebra from "Schaum's Outline of Linear Algebra." It's cheap, it's thorough, and it shows its work in calculations. But it's also *too* thorough (it covers a lot of stuff you won't need) and more abstract than intuitive. Schaum's is a lot better than nothing, but it might not be the best for you.
asen7
#16
Apr26-11, 04:30 PM
P: 13
When I was going over the topic, I found something about QED and QCD which seem interesting. But how is QCD and QED different from quantum mechanics, and what can you do with QED and QCD?

If I wanted to focus in more on these two branches, would I need to learn any more levels math other than the ones listed above?

Thank you very much, I really appreciate the responses so far.


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