What Is the Role of the Hamiltonian in Quantum Mechanics?

[Xezlec]

After several failures in the past (why does the universe have to be so complicated?!), I'm once again trying to learn to understand the basics of QM, out of sheer frustration with not knowing what the heck physicists are talking about all the time. I know, I still have a long way to go.

Anyway, I just started again and I'm already confused about something.

I'm led to understand that the rules governing the time evolution of the quantum state, together with the definitions of the observable quantities (in the form of their associated operators) take the place of Newton's laws of motion and the classical definitions of the quantities. Is that about right?

If so, why can't I find a list of the definitions of the operators of all the usual physics quantities written somewhere? I've found the operators for momentum, position, and spin, and they seem to make sense to me (with what little I know), but I can't find the definition of the Hamiltonian, which is needed for me to know how energy is defined in quantumland. More importantly, since the entire rules for how stuff happens are encoded in the Schrödinger equation, which relies on the undefined Hamiltonian, I can't imagine (or compute) how anything happens at all.

Surely it has a definition somewhere? I mean, it can't just be that I make one up... thereby making up any laws of motion I want for my universe...

Thanks, and apologies for my utter n00bity.

 

[dx]

The energy of a particle in quantum mechanics is the same as in classical mechanics :

$$H = \frac{p^2}{2m} + V(x)$$.

Just replace the classical quantities with the respective operators, and you get the hamiltonian operator.

 

[cmos]

why can't I find a list of the definitions of the operators of all the usual physics quantities written somewhere? I've found the operators for momentum, position, and spin, and they seem to make sense to me (with what little I know), but I can't find the definition of the Hamiltonian, which is needed for me to know how energy is defined in quantumland. More importantly, since the entire rules for how stuff happens are encoded in the Schrödinger equation, which relies on the undefined Hamiltonian, I can't imagine (or compute) how anything happens at all.

All classical dynamical variables may be defined in terms of position and momentum. Therefore you have already found what you need.

Regarding the Hamiltonian...

Surely it has a definition somewhere? I mean, it can't just be that I make one up... thereby making up any laws of motion I want for my universe...

dx has shown you the general form of the Hamiltonian, where the first term is simply the kinetic energy (again, define in terms of momentum as mentioned above) and the second term is the potential. In a sense, you do get to "make up" your Hamiltonian. The potential is determined by you, God, and/or may be inherent to your system.

For example, in the case of atomic hydrogen (one proton, one electron) the potential is simply the Coulomb potential. If you would like to set up your own crazy experiment with a charged particle placed off-center in an electric octupole, then you will have a more involved potential making for a more involved Hamiltonian.

 

[Xezlec]

Thanks. But how/why can I replace quantities with operators? As I understood it, the relationship between a quantity and the operator is complicated and indirect: the quantity lives in the eigenvalues of the operator. I wouldn't expect to be able to add two operators in general and get the operator corresponding to a quantity that is the sum of the quantities represented by those two operators... would I? *brain smokes*

And I guess the operator for that potential V(x) is just the operator that multiplies V(x) by phi(x) for all x... that appears to work out right.

 

[cmos]

Not exactly sure what you mean by "quantities." Physical observables are represented by operators. All of your classical observables may be defined by position and momentum, so that's all you need (spin is another story, but that doesn't have a classical counterpart).

What you might be getting tripped up on is the fact that operating the Hamiltonian on the wavefunction yields an eigenvalue equation. Well this is only true when you take out time dependence, hence the time-independent Schrodinger equation. Well, this is actually DERIVED from the more general time-dependent Schrodinger equation. From this derivation we see that the time-independent wavefunctions are actually energy eigenfunctions.

The same does not hold true for either momentum or position. The reason for this is that you may not have a determinant state of either momentum or position (a la Heisenberg). So your eigenvalue equations will not exactly behave in the simple manner as we have discussed above.

 

[pmb_phy]

Thanks. But how/why can I replace quantities with operators?

I believe that this is not something that can be proved through an argument or derivation. I think that this is embedded in the postulates of quantum mechanics.

As I understood it, the relationship between a quantity and the operator is complicated and indirect: the quantity lives in the eigenvalues of the operator. I wouldn't expect to be able to add two operators in general and get the operator corresponding to a quantity that is the sum of the quantities represented by those two operators... would I? *brain smokes*

Only if there is a corresponding classical equivalent.

And I guess the operator for that potential V(x) is just the operator that multiplies V(x) by phi(x) for all x... that appears to work out right.

The x in V(x) is replaced with the operator X. This operator is defined such that X|$\Psi$> = x|$\Psi$>. V(X) is defined by its series expansion in the operator X.

Pete

 

[Xezlec]

I was trying to put together another response, but I think my understanding of math is just not up to the task, which I think points to the fundamental problem. Maybe I still don't even understand the framework of the basic formulation of the theory. Can anyone recommend maybe what kinds of mathematics I should study to learn enough to approach basic QM? Any recommendations for URLs or, less desirably, books that might be a good first step?

 

[dx]

You can study elementary QM using just the Schrodinger equation if you know some basic calculus and differential equations. If you want to study the full theory of QM you need to know at least advanced calculus and linear algebra (and also a good amount of physics to appreciate it fully).

 

[Xezlec]

OK, well I think my vector cal is pretty good and my linear algebra is decent and improving, so I would think I should be able to understand this. But I guess I'm not getting the concept of an observable or an operator or something because so far it isn't making sense and I'm not even effectively communicating what the problem is.

 

[dx]

If you're comfortable with vector calculus and linear algebra, you should be able to start learning quantum mechanics.

In classical mechanics, systems have states which are points in a set. For example, a particle on a line has a position which is an element of R. The observables in classical mechanics are basically just the states, or numbers representing the states. In quantum mechanics, these points in sets are replaced by vectors in a vector space. The observables of quantum mechanics are not the states, but operators on this vector space. It's normal to feel a little uncomfortable with this at the beginning, but I suggest you not worry about this and go ahead and learn quantum mechanics. You can worry about what exactly it means and how you interpret it after you learn it.

 

[Xezlec]

Wait, you're saying the observable somehow *is* the operator? I don't even get wat that means... I need a better book.

 

[Xezlec]

You know what, never mind. I'm chalking this up as another abortive attempt. I'll wait a few years and try again. Sorry about this.

 

[dx]

Wait, you're saying the observable somehow *is* the operator? I don't even get wat that means... I need a better book.

No I'm sorry, I wasn't clear. Let's consider a particle moving on a line. In classical mechanics, it is assumed that at every instant of time t, the particle has a certain position on the line x. That is it's state at time t. In quantum mechanics, the thing that is known at every instant of time is not its position, but something else. This is the state vector. This is a complex function defined on the line: $$|\psi(x)>$$. That is the analog of the classical state.

Now I will give a certain interpretation, which is not accepted by everyone. You can think of this state as somehow representing your knowledge about the particle. This is loosely just a probability density on the line. Now there are certain states which correspond to a certain knowledge that the particle is at a particular position. These are the eigenstates of the position operator $$\hat{X}$$. In the case of a particle moving on the line, they are the Dirac delta functions. There are also momentum eigenstates, which are "states of knowledge" where the momentum is exactly known. These are the eigenvectors of the momentum operator which are

$$e^{i \frac{p}{h}x}$$.

If your "state of knowledge" is one of the momentum eigenstates, then it means that you know certainly what the momentum will be when you measure it.

 

[comote]

Given a classical observable f(q,p) there is a self adjoint operator F(Q,P) acting on Hilbert space that represents the expected outcomes.

For simple cases
$$H = \frac{p^2}{2m} + V(q)$$.
this is easily understood. Unfortunately what to do for the
$$q^np^m$$
terms is a little more complicated and not fully understood. A common formula is to transform
$$qp\to\frac{1}{2}(QP+PQ)$$.

 

[dx]

You know what, never mind. I'm chalking this up as another abortive attempt. I'll wait a few years and try again. Sorry about this.

No! don't give up. Quantum mechanics is fascinating. Maybe you're just not reading the right books or something. I suggest the first few chapters of Feynman lectures vol 3 for an introduction. Also, Leonard Susskind (one of the founders of string theory) has a excellent set of video lectures on quantum theory you can find on youtube. Then try J. J. Sakurai's book if you are confident about your maths. Otherwise, try Shankar or Griffiths.

 

[Fredrik]

If so, why can't I find a list of the definitions of the operators of all the usual physics quantities written somewhere? I've found the operators for momentum, position, and spin, and they seem to make sense to me (with what little I know), but I can't find the definition of the Hamiltonian, which is needed for me to know how energy is defined in quantumland. More importantly, since the entire rules for how stuff happens are encoded in the Schrödinger equation, which relies on the undefined Hamiltonian, I can't imagine (or compute) how anything happens at all.

One of the postulates of QM is that states are represented by vectors in some vector space. A vector is often written using the "ket" notation: $|\alpha\rangle$. If you describe the system as being in state $|\alpha\rangle$, then an observer who's translated in time relative to you would describe the system as being in another state $|\alpha'\rangle$. There must be an operator U(t) that takes $|\alpha\rangle$ to $|\alpha'\rangle$, and it must be an exponential because it has to satisfy U(t+t')=U(t)U(t'), so we can write it as $U(t)=e^{At}$, where A is some operator. (Its existence is guaranteed by the condition U(t+t')=U(t)U(t')). It's convenient to choose U(t) to be unitary so that it doesn't change the norm of the state vectors it acts on. The requirement that U(t) be unitary implies that A must be anti-hermitian (i.e. $A^\dagger=-A$). We prefer to work with hermitian operators because their eigenvalues are real, so let's define H=iA. H is now hermitian, and $U(t)=e^{-iHt}$.

That's the definition of the Hamiltonian. It's the generator of translations in time. The momentum operators can be defined the same way as generators of translations in space, and the spin operators can be defined as generators of rotations.

 

[Fredrik]

Can anyone recommend maybe what kinds of mathematics I should study to learn enough to approach basic QM?

Linear algebra is by far the most important kind, but you can probably just study those parts you need when you run into difficulties in your QM book. You obviously have to understand complex numbers too.

Wait, you're saying the observable somehow *is* the operator? I don't even get wat that means... I need a better book.

Yes, an observable is an operator.

 

[Xezlec]

OK, let's keep going I guess. Most of what's being said here are things I already understand, or at least think I understand. It's the details that are confusing me.

Let me try stating the path I've followed and see if there are any objections. Maybe this will let me debug my brain glitch.

1. The state of a particle is described by a set of complex numbers called a wavefunction. From one point of view, there can be said to be N complex numbers for each point in space, where N is the number of legal spin states. In this representation (the position/spin eigenstate basis) the probability of observing the particle with a particular combination of spin and position equals the squared magnitude of the complex number for that particular combination. Spin and position are therefore two quantities that the wavefunction tells us something about.

2. The procedure for extracting information about other quantities (that is, quantities other than position and spin, which were very convenient in this basis) is more involved. To find out what the probability is of measuring the system to have a value p for a quantity (say, momentum), I have to solve the equation P Phi = p Phi, where Phi is an unknown vector of zillions of complex numbers and P is the operator corresponding to the quantity in question (momentum in this case). The result will be a wavefunction Phi that is the eigenstate of the system for the momentum value p. So now I take the inner product of Phi with the actual wavefunction and the result will be the probability of measuring the particle to have a momentum of p.

*catches breath*

3. Now that our wavefunction has some defined connection to measurable quantities, we have to find the rules for how wavefunctions behave. These rules are defined by saying that every wavefunction is a superposition of energy eigenstates (obviously). In each of those eigenstates, the components all keep the same magnitude with time, but their phases all spin around (counterclockwise!) at a frequency proportional to the value of energy for this eigenstate (that is, the eigenvalue).

4. But wait! This doesn't help us yet! In order for that to have any bearing on what happens to the observable quantities in time, we need to know what the energy eigenstates look like in terms of the basis we were using at the beginning (the position/spin eigenstate basis), and that can only be done if we know the energy operator so we can solve an equation as in #2. So apparently the total energy operator is related to the momentum operator the same way that the energy quantity/eigenvalue/<insert jargon here> is related to the momentum <whatever>. And that is true because...?

 

[Domenicaccio]

Thanks. But how/why can I replace quantities with operators? As I understood it, the relationship between a quantity and the operator is complicated and indirect: the quantity lives in the eigenvalues of the operator. I wouldn't expect to be able to add two operators in general and get the operator corresponding to a quantity that is the sum of the quantities represented by those two operators... would I? *brain smokes*

And I guess the operator for that potential V(x) is just the operator that multiplies V(x) by phi(x) for all x... that appears to work out right.

I have vague memories but...

1) express the quantity/observable in terms of coordinates x, y, z and momentum $$p_x$$, $$p_y$$, $$p_z$$

2) the operator corresponding to a simple coordinate is the multiplication by such coordinate

3) the operator corresponding to a momentum is $$-i \frac{h}{2\pi} \frac{\delta}{\delta x}$$



Therefore for example, if you need the operator for kinetic energy

$$T=\frac{p_x^2+p_y^2+p_z^2}{2m}$$

then you substitute each p with the corresponding

$$-i \frac{h}{2\pi} \frac{\delta}{\delta x}$$,

square it to

$$-\frac{h^2}{4\pi^2} \frac{\delta^2}{\delta x^2}$$

and the final operator will be

$$-\frac{h^2}{8m\pi^2} \nabla^2$$

 

[Xezlec]

Therefore for example, if you need the operator for kinetic energy

$$T=\frac{p_x^2+p_y^2+p_z^2}{2m}$$

then you substitute each p with the corresponding

$$-i \frac{h}{2\pi} \frac{\delta}{\delta x}$$,

square it to

$$-\frac{h^2}{4\pi^2} \frac{\delta^2}{\delta x^2}$$

and the final operator will be

$$-\frac{h^2}{8m\pi^2} \nabla^2$$

Interesting. That derivation appears to implicitly assume that an operator applied twice corresponds to a quantity that is the square of the quantity represented by the original operator. Add that to the list of fascinating and unintuitive properties of quantum operators that have yet to be explained to me in any form I can understand. The other thing on the list right now is that adding two operators produces the operator corresponding to the quantity that is the sum of the quantites represented by the original two operators.

If there is a trend here, it sounds like it would be that whatever you do to the operators will always "look the same" notationally as whatever happens to the quantities. I wonder how this mysterious connection is achieved, and how far it goes. I'm still thinking about it. Maybe this new information will help me come up with the answer.

I'm trying to gather enough information that I can infer a full list of the basic postulates of QM. That way, I will be able to understand everything else based on how it is derived from those postulates. I don't know yet whether this particular question is related to some more postulates that I don't know, but it sounds interesting enough that it might be.

 

[Xezlec]

OK, I can see why both of the above magical properties of operators are true if it is possible for the system to be in eigenstates of both operators at once. I do not, however, see any particular reason why they are true for operators for which this is not possible. So let me ask another question:

Do the postulates of QM include the postulates of Newtonian mechanics, or can the postulates of NM be viewed as derived from QM?

 

[nrqed]

Interesting. That derivation appears to implicitly assume that an operator applied twice corresponds to a quantity that is the square of the quantity represented by the original operator. Add that to the list of fascinating and unintuitive properties of quantum operators that have yet to be explained to me in any form I can understand. The other thing on the list right now is that adding two operators produces the operator corresponding to the quantity that is the sum of the quantites represented by the original two operators.

If there is a trend here, it sounds like it would be that whatever you do to the operators will always "look the same" notationally as whatever happens to the quantities. I wonder how this mysterious connection is achieved, and how far it goes. I'm still thinking about it. Maybe this new information will help me come up with the answer.

I'm trying to gather enough information that I can infer a full list of the basic postulates of QM. That way, I will be able to understand everything else based on how it is derived from those postulates. I don't know yet whether this particular question is related to some more postulates that I don't know, but it sounds interesting enough that it might be.

QM is a deep subject and there are many ways to introduce it. There are many layers of subtleties. Some people like to jump into the most general (and most abstract) formalism. Others prefer to introduce the subject more gently, not worrying about being as general as possible to start with, knowing that with time the student will eventually reach the point where he/she will be ready for the next layer of subtlety.

The most basic approach is what some have mentioned: take the classical expression for the Hamiltonian (NOTE: for some systems, there is NO classical hamiltonian to use as a starting point in which case you must use other principles to guide you. This is the case with spin interactions which have no classical analogue). But let's say you have a classical Hamiltonian to work with. Then simply leave x,y,z as the positions and replace $$p_x, p_y, p_z$$ by their operator expression $$- i \hbar \frac{\partial}{\partial_x}, \ldots$$ and so on. Usually this is straightforward (there could be so-called ordering ambiguities in some exceptional cases but no need to worry about this for now).

Then solve Schrodinger's equation to determine the time evolution of the wavefunction.
You must pick an initial state that satisfied the boundary conditions you have and which is square integrable. Once you have the wavefunction at any time you can answer any physical question like: if I measure the angular momentum at a specific time, what are the possible results I can get and with what probabilities?).

That's all there is to it.

When you include spin, things are a bit more subtle.

 

[Xezlec]

QM is a deep subject and there are many ways to introduce it. There are many layers of subtleties. Some people like to jump into the most general (and most abstract) formalism. Others prefer to introduce the subject more gently, not worrying about being as general as possible to start with, knowing that with time the student will eventually reach the point where he/she will be ready for the next layer of subtlety.

I think it's the student that makes the difference between these approaches. I, for instance, am a detail thinker. I am not often capable of understanding or working with "high-level overviews" or partial descriptions of systems. This has been a big disadvantage for me in a world expressly designed for the other kind of person. I always had trouble reading block diagrams of circuits until I learned to visualize specific gate-level implementations of each block in the picture and mentally superimpose them on top of the blocks. What I am best at is instantly seeing every possible consequence of a system of a small number of absolutely precise, rigid, unambiguous rules.

I feel like the moment I comprehend that very last rule, I will immediately see everything. Thousands and thousands of true statements will quickly rush into mind. The moment before I understand that last rule, I will still see nothing.

The most basic approach is what some have mentioned: take the classical expression for the Hamiltonian (NOTE: for some systems, there is NO classical hamiltonian to use as a starting point in which case you must use other principles to guide you. This is the case with spin interactions which have no classical analogue). But let's say you have a classical Hamiltonian to work with. Then simply leave x,y,z as the positions and replace $$p_x, p_y, p_z$$ by their operator expression $$- i \hbar \frac{\partial}{\partial_x}, \ldots$$ and so on. Usually this is straightforward (there could be so-called ordering ambiguities in some exceptional cases but no need to worry about this for now).

Then solve Schrodinger's equation to determine the time evolution of the wavefunction.
You must pick an initial state that satisfied the boundary conditions you have and which is square integrable. Once you have the wavefunction at any time you can answer any physical question like: if I measure the angular momentum at a specific time, what are the possible results I can get and with what probabilities?).

That's all there is to it.

My lack of a roaring blast of sudden realizations suggests to me that is not the case.

I don't intuitively see what the rules are for constructing an equation of operators based on an equation of classical variables. I have gathered from other posts that application of an operator is the equivalent of multiplication by a classical quantity, and addition is addition, but these two facts by no means suggest to me the general character of this weird transformation. What is the operator equivalent of integration of a classical quantity, for example? What about exponentiation? Sinusoids? More importantly, how would I figure out what one of these were if I didn't already know?

On top of that, I am not convinced that this procedure preserves the relationships I require to be able to think of this as a transformation of ordinary arithmetic. For example, I can't see how commutativity of multiplication is preserved when it turns into operator application.

I'm sorry, but, your valiant efforts notwithstanding, so far I still understand nothing. :-(

When you include spin, things are a bit more subtle.

As I've mentioned already, spin is the only aspect of quantum physics I do understand.

 

[malawi_glenn]

Xezlec: Just start with Sakurai modern quantum mechanics. Start on page 1, and go from page to page, and try to fill in the steps.

The first chapter starts with the postulates of QM, states and hilbert space, commutators and observables vs. operators and momentum and position.

The Second chapter deals with time evolution (dynamics), and it is here that the Hamiltonian will appear since it is, as Fredrik pointed out, the generator of time evolution.

The third chapter deals with angular momentum and spin in quantum mechanics.

My explanation to operator:
And operator is a general description of an 'action' you perform on a function (state in QM). An operator can be just a number, or a derivative, a second order derivative, or a combination of those.

The quantum mechanical operators relation to their classical identity dependend on many things, like what representation you have and so on. That are quite advanced topics, which you DON'T have to learn from the beginning in order to appreciate and study QM. Also, one needs quite solid knowledge of classical mechanics in order to 'obtain' their quantum mechanical counterparts.

http://hyperphysics.phy-astr.gsu.edu/Hbase/quantum/qm2.html#c1

http://hyperphysics.phy-astr.gsu.edu/Hbase/quantum/qmoper.html#c1


As a general example:

Consider state $$|\alpha > = |\vec{x} = (x',y',z'), S_z = + >$$

Now the primes are just labels, they don't symbolize anything with derivatives, it is a common thing, that one let's prime also denote other things than order of derivation.

consider an action of operator $$\hat{x} |\alpha > = x'|\alpha >$$

and

$$\hat{S_z}|\alpha > = +(1/2)\hbar|\alpha >$$


What do you mean by:
"What is the operator equivalent of integration of a classical quantity, for example?"

It is a pretty vauge question, can you clarify? You mean like integrating the kinetical energy of a system over time?

 

[nrqed]

On top of that, I am not convinced that this procedure preserves the relationships I require to be able to think of this as a transformation of ordinary arithmetic. For example, I can't see how commutativity of multiplication is preserved when it turns into operator application.

Commutativity of multiplication is in general not [/i] preserved when the classical quantities are replaced by operators. For example replacing XP by operetaors and replacing PX by operators does not give the same result even though the classical expressions are obviously equal (this is a so-called ordering ambiguity). This is because the operators X_op and P_op do not commute.

You know, I had the same frustrating feeling about superstring theory a few years ago. I kept reading and feeling more and more frustrated because I could not understand the BIG picture and see how everything felt together. It seemed as if I had to master supersymmetry, conformal field theory, quantum field theory, topology, differential geometry, etc all at once before I could understand string theory! I was getting very frustrated.

And then I realized that I should not worry about getting everything right away. I simply had to keep reading and to understand little bits by little bits. The trick was to keep reading different books and papers and sometimes to reread them again and again. At first I would understand only 5% of what a paper said. Then I would understand some other pieces from other papers and books, then I would come back and 15% of the same paper now made sense. And I would go back and read otherpapers and books. It was frustrating to read stuff and to not understand most of what was written, even papers who claimed to be pedagogical introduction to the topic. The key point was to not let myself stay stuck on papers that did not make sense to me. I would read them but not try to understand everything at first. It's only after a year of understanding tons of little details that suddenly I reached a critical mass of understanding that was enough to finally start to be able to learn from textbooks. Once you reach a critical mass, there is snowball effect, the more you understand, the easier it is to understand new aspects. But the key point is to reach the critical mass and during that initial period, the key point is not to worry if everything does not fit into place right away! Otherwise there is the danger of getting discouraged and frustrated and learning is no longer fun.

But the good thing about QM i sthat there are many excellent textbooks on it so I would suggest you pick a good one and just work your way throug it, aksing questions here if you are stuck. I would suggest Shankar.

 

[muppet]

Richard Feynman once said that he could safely say that no-one understood quantum mechanics... and he got a Nobel prize for reformulating it! So DO NOT give up if it seems confusing... it is confusing :tongue:
The thing it sounds like is really confusing you at the minute is how physical quantities relate to the wavefunction. First of all, forget about manipulating them like you would some scalar algebraic letter; don't worry about "exponentiating" or "taking the sine of" or anything like that. Work out what you actually should do to them first. As has already been mentioned, a lot of these properties don't translate across; in particular, the operators for position and momentum (for example) do not commute the way classical scalars would. (And when you understand why, you really will understand QM!)
For the first understandings of how you relate the wavefunction to reality, you might find it helpful to think about the historical development of the subject. Planck's oscillators and Einstein's photoelectric effect linked the energy of a wave to its frequency; shortly afterwards, De Broglie proposed that a particle had a wavelength, linked to its momentum. But suppose you already had a wave, and wanted to work out a momentum from that. How would you do it?
Say you've described your 'wave' by a complex exponential:
$$\psi= e^{i(kx-\omega.t)}$$
You could read off k (wavenumber; = 2pi/wavelength) straight away, and use De Broglie's formula linking wavelength and momentum:
$$\lambda=\frac{h}{p}$$
There is a problem with this approach, however, which is that in general you're trying to work out what the wavefunction is in the first place! So you have to be slightly cleverer than that. If you were in fact being terribly clever (clever enough to win a Nobel prize...) you could spot that if you take such a function and differentiate it, you get i times the physical quantity you're interested in, k, multiplied by the function. Now i is easily gotten rid of- you just multiply by -i to get 1. So all of a sudden you have differential equation to solve! :
$$-i\hbar\frac{d}{dx}\Psi=p\Psi$$
where the h-bar symbol is Planck's constant h divided by pi, which I'd guess you've already come across- useful for getting rid of factors of 2pi picked up when you're dealing with waves!
The first thing is to notice is that this is an eigenvalue equation. You operate on a vector, you get the vector back multiplied by some scalar. QM postulates that the result of every physical measurement is an eigenvalue of some operator. This is why we associate the differential operator above with momentum, because all momentum measurements are eigenvalues of this operator.
The second thing to do is to see what happens when we proceed by analogy with classical mechanics. The classical formula for the total energy of a particle is
$$\frac{p^{2}}{2m}+V(x,t)=E$$
where V is the potential experienced by the particle, say due to gravity or an electric field. Replacing multiplication by p by the differential operator above the left hand side becomes:
$$(\frac{\hbar^2}{2m}\frac{d}{dx}+V(x))\Psi$$
Exactly the same way as we did for de Broglie's equation above, using E=hf and our complex exponential, you can work out a differential operator for energy. The end result being (drum roll...)
$$(\frac{\hbar^2}{2m}\frac{d}{dx}+V(x,t))\Psi=-i\hbar\frac{d}{dt}\Psi$$
and you have 'derived' (sort of) the Schroedinger equation!
...
That was a long post, but I hope it helped!

 

[Mentz114]

Xezlec,

I'm sorry, but, your valiant efforts notwithstanding, so far I still understand nothing. :-(

I think you do yourself an injustice. Anyway, I'd like to introduce the Pauli method, in case you haven't come across it. If this is all old hat to you, well, too bad.

Wolfgang Pauli taught QM and used the analogy between matrices and operators. Starting with the quantum linear harmonic oscillator he illustrates the relevant operators like this. The basic equation for the QHO can be reduced to

$$H = \frac{h\omega_0}{2}\left(X^2 + P^2)$$

where H, X and P are the Hamiltonian, the position and the momentum. This equation is true for classical quantities and we'd like it to be true for the quantum case also.

We can now define some matrices -

$$X = \frac{1}{\sqrt{2}}\[ \left[ \begin{array}{ccccc} 0 & \sqrt{1} & 0 & 0 & ... \\\ \sqrt{1} & 0 & \sqrt{2} & 0 & ... \\\ 0 & \sqrt{2} & 0 & \sqrt{3} & ... \\\ 0 & 0 & \sqrt{3} & 0 & ... \\\ ... & ... & ... & ... & ... \end{array} \right]\]$$


$$P = \frac{1}{i\sqrt{2}}\[ \left[ \begin{array}{ccccc} 0 & \sqrt{1} & 0 & 0 & ... \\\ -\sqrt{1} & 0 & \sqrt{2} & 0 & ... \\\ 0 & -\sqrt{2} & 0 & \sqrt{3} & ... \\\ 0 & 0 & -\sqrt{3} & 0 & ... \\\ ... & ... & ... & ... & ... \end{array} \right]\]$$

If we plug these into the expression for H we get

$$H = h\omega_0\[ \left[ \begin{array}{ccccc} \frac{1}{2} & 0 & 0 & 0 & ... \\\ 0 & \frac{3}{2} & 0 & 0 & ... \\\ 0 & 0 & \frac{5}{2} & 0 & ... \\\ 0 & 0 & 0 & \frac{7}{2} & ... \\\ ... & ... & ... & ... & ... \end{array} \right]\]$$

These matrices behave exactly like the operators in this discrete state space. You can check the commutation relations are correct, the expectations are correct, and as you can see the Hamiltonian operator has all the right eigenvalues. I'll leave you to figure out the state vectors.

Now I've just pulled these out the hat, but Pauli does give the full justification in 'Wave Mechanics' ( reprinted by Dover as vol 5 of the 'Pauli Lectures' set. Good value).

M

 

[Xezlec]

So far I seem to have failed to communicate either my question or the abuse of notation I keep seeing, and most people who respond still seem not to notice the notation problem and just take it for granted. The responses I'm getting are either restating simple facts I already explained that I understood in earlier posts, or telling me to read various textbooks. The latter suggestion is quite reasonable, and I've noted the recommended titles for a possible trip to the bookstore later. But I still think the simple, subtle axiom I'm looking for can probably be given in under 10 words if I can just communicate my question, and maybe even just with a single equation. So, before I give up entirely on this thread, I will try one more time to explain in a different way.

I believe I can get closer to an answer to my main question if the following 2 questions can be answered precisely. Emphasis on "precisely". Formulas and equations are tremendously preferred over words.

1. What is the formula for the probability distribution of a quantity that is momentum times position, given an arbitrary wavefunction $$\Psi$$? Or does QM simply state that no such quantity can be measured or has any meaning?

2. What is the formula for the probability distribution of a quantity defined by $$e^{p \over p_0} + {x \over x_0}$$, given an arbitrary wavefunction $$\Psi$$? Or does QM simply state that no such quantity can be measured or has any meaning?

As far as I know, this one issue (exactly how to map the algebra formula for the classical quantity you're talking about to the operator formula for the quantum operator you need) is the only fundamental thing I don't get right now. I just cannot bring myself accept that the answer is "pick whatever operator formula looks the most visually similar to the algebra formula when written out in standard mathematical notation and pray it's right".

Also, I'm now trying to learn to use TeX, since everyone else seems to use it religiously. Wish me luck. $$O O \over U$$

 

[Xezlec]

OK never mind about #2, apparently I'm too dumb to know simple math when I'm sitting here pondering this stuff in the wee hours of the morning. The answer to #2 obviously follows immediately from the rule for multiplication and the rule for addition, you just have to use a trivial Fourier transform. Geez I feel dumb for missing that for like a week...

But I still don't see #1 yet.

 

[Mentz114]

This is your original post -

After several failures in the past (why does the universe have to be so complicated?!), I'm once again trying to learn to understand the basics of QM, out of sheer frustration with not knowing what the heck physicists are talking about all the time. I know, I still have a long way to go.

Anyway, I just started again and I'm already confused about something.

I'm led to understand that the rules governing the time evolution of the quantum state, together with the definitions of the observable quantities (in the form of their associated operators) take the place of Newton's laws of motion and the classical definitions of the quantities. Is that about right?

If so, why can't I find a list of the definitions of the operators of all the usual physics quantities written somewhere? I've found the operators for momentum, position, and spin, and they seem to make sense to me (with what little I know), but I can't find the definition of the Hamiltonian, which is needed for me to know how energy is defined in quantumland. More importantly, since the entire rules for how stuff happens are encoded in the Schrödinger equation, which relies on the undefined Hamiltonian, I can't imagine (or compute) how anything happens at all.

Surely it has a definition somewhere? I mean, it can't just be that I make one up... thereby making up any laws of motion I want for my universe...

Thanks, and apologies for my utter n00bity.

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

So far I seem to have failed to communicate either my question or the abuse of notation I keep seeing, and most people who respond still seem not to notice the notation problem and just take it for granted. The responses I'm getting are either restating simple facts I already explained that I understood in earlier posts, or telling me to read various textbooks. The latter suggestion is quite reasonable, and I've noted the recommended titles for a possible trip to the bookstore later. But I still think the simple, subtle axiom I'm looking for can probably be given in under 10 words if I can just communicate my question, and maybe even just with a single equation. So, before I give up entirely on this thread, I will try one more time to explain in a different way.

I for one will not be attempting to help you again.

M

 

[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