Quantum Mechanics to a Newcomer

[Xyius]

Hello all, I was wondering if I could get some input from those who know Quantum Mechanics and may be able to help me with my understanding.

I am taking my first Quantum Mechanics course. It is a graduate level course (I am at the end of my Junior year as an undergrad) and I have not taken any introduction courses to Quantum Mechanics before. I am about a month in and I am doing fine in it so far. Although I am doing fine, I still have some confusion about some main concepts. This past month has been entirely mathematics with some examples of spin 1/2 systems thrown in. At my level of understanding it all seems extremely abstract. My main problem is relating it to real life scenarios.

For example, I know that an observable is measurements made in real life to some quantity and when we take these measurements, the system is thrown into an eigenstate of the observable. But this concept of observable still seems very ambiguous to me. How would one obtain the information on an observable? And by that I mean, how would these "measurements" translate into a matrix?

Right now I am being thrown so much mathematics, and it is hard to take it all in without some grip on how to apply it or where I could use it.

My other question is, where did the Schrodinger equation come from? My professor says it is a fundamental postulate of quantum mechanics and cannot be derived, but what was the logic used to construct it?

Thanks

 

[atyy]

The Schroedinger equation should be taken as postulate, like F=ma.

But for motivation you can think of the Bohr model of the atom, and its successful predictions. Part of the model is a wave. The Schroedinger equation was a search for a wave equation that would give the Bohr wave as its solution. Initially Schroedinger tried the Klein-Gordon equation, but that didn't work.

 

[Xyius]

The Schroedinger equation should be taken as postulate, like F=ma.

But for motivation you can think of the Bohr model of the atom, and its successful predictions. Part of the model is a wave. The Schroedinger equation was a search for a wave equation that would give the Bohr wave as its solution. Initially Schroedinger tried the Klein-Gordon equation, but that didn't work.

That makes sense but I am looking for something a little stronger. My view is, if I was Schrodinger, how would I construct this equation? What logic would draw me to the final form of this equation. Was there experimental motivation such as in F=ma?

 

[jtbell]

where did the Schrodinger equation come from?

https://www.physicsforums.com/showthread.php?p=418069#post418069

 

[strangerep]

[...] if I was Schrodinger, how would I construct this equation? What logic would draw me to the final form of this equation. Was there experimental motivation such as in F=ma?

Sometimes it's better to follow a more modern approach, rather than retrace the more convoluted older methods...

The Schrodinger equation need not be taken as an axiom, since it's just a particular case of the relationship between the Hamiltonian of a system and its time-evolution. Since QM deals with probability distributions for the dynamical quantities characterizing a particular (class of) system, much of the derivation of ordinary QM boils down to ensuring that expectations values of these quantities behave sensibly under the group of Galilean transformations (spatial rotations & translations, velocity changes, time translation) which are reasonably familiar in everyday life. The dynamical equations of QM are then a direct consequence of requiring that our mathematical model of the (statistical) experiments we can perform should be consistent with such boringly familiar transformations. (This is the so-called "group theoretic" approach to the derivation of quantum theory, and is much more general and compelling than older methods.)

You didn't say which textbook(s) were recommended for your course, but since it's a graduate level course, I'll say that Ballentine's "QM -- A Modern Development" explains the above quite well, motivating the abstract from the real world. IMHO, it should be a standard text in modern QM courses. :-)

 

[ZapperZ]

I am taking my first Quantum Mechanics course. It is a graduate level course (I am at the end of my Junior year as an undergrad) and I have not taken any introduction courses to Quantum Mechanics before.

Holy cow! What school is this that would allow you to take a graduate level QM without having taken any QM courses previously?

No wonder you're struggling with the math! I see a lot of pain ahead.

Zz.

 

[Xyius]

You didn't say which textbook(s) were recommended for your course, but since it's a graduate level course, I'll say that Ballentine's "QM -- A Modern Development" explains the above quite well, motivating the abstract from the real world. IMHO, it should be a standard text in modern QM courses. :-)

I am using "Modern Quantum Mechanics" by Sakurai.

Holy cow! What school is this that would allow you to take a graduate level QM without having taken any QM courses previously?

No wonder you're struggling with the math! I see a lot of pain ahead.

Haha yeah I know its kind of insane. I am at Stevens Institute of Technology. The funny thing is, is I actually did it by mistake. I saw "Quantum Mechanics I" and figured "well hey, this must be the first course in QM." And it allowed me to sign up. I found out recently that there is a course called "Quantum Mechanics and Engineering Applications" for undergrads that I was SUPPOSED to go in.

The thing is, is there has not been a point where I felt helplessly lost. I am doing fine in the course so far, but I can only imagine that it gets worse and you are probably right about the pain :p

Thanks everyone for your responses so far. It has really helped me.

 

[jewbinson]

You should tell your lecturer the situation you're in and ask him if you'll survive (the course) if you carry on. He will know the answer better than anybody.

 

[jewbinson]

Edit: and if not, you should opt out and go for the engineering/beginners one one

 

[genericusrnme]

My other question is, where did the Schrodinger equation come from? My professor says it is a fundamental postulate of quantum mechanics and cannot be derived, but what was the logic used to construct it?

I'd say this was partially untrue..
You CAN derive schrodingers equation by taking the quantities related to time translation (Hamiltonian) and using them in your time evolution operator, this gives you schrodinger's time dependant equation. The 'time independant' equation is just an eigenvalue equatuation.
Unless he meant the whole observables -> operators thing then yes, that's a postulate and you've got to learn to deal with it :p

For example, I know that an observable is measurements made in real life to some quantity and when we take these measurements, the system is thrown into an eigenstate of the observable. But this concept of observable still seems very ambiguous to me. How would one obtain the information on an observable? And by that I mean, how would these "measurements" translate into a matrix?

You see how they act on basis kets and you derive them as generators of something for the most part. The Hamiltonian - time translation, Momentum - position translation, Angular Momentum - rotations.

At my level of understanding it all seems extremely abstract.

Yeah, it's probably going to feel like that 90% of the time :L

I am using "Modern Quantum Mechanics" by Sakurai.

This is a very fine book!
You might also want to check out Landau and Lifgarbagez book on non relatavistic QM also, it might help solidate some ideas, seeing them from a different perspective. I found some parts of the L&L book very insightful, albeit a bit older and less (virtually none) dirac notation.

But for the love of god, though, don't go near Griffith's Intro to QM book.
It'll leave you feeling like you know nothing because the book leaves you completely unprepared for the problems it gives you and the problems are ridiculously hard for the most part.
That and it's nowhere near graduate level.


I'd definitely talk to the lecturer about it though. He'll know what's in store with the course, it could be that there's a load of mathematics now and you'll get one piece of information and everything will click, or it could be the case that things are going to get harder and more abstract with little room for that 'click'...