[Homework Help] Quantum Physics I

[Bears!]

Hi all,

So I'm just starting QM this quarter and expect to need help regularly (it's the first undergraduate QM class, so it's for noobs). I don't think it's against the rules to keep a homework help thread going (so long as I let you guys know when I finished a problem). I just don't want to be obnoxious about always asking questions is all! Anyhow, I'll certainly be needing hints and pointers throughout the quarter since I currently have zero friends in my classes as a result of transferring programs.

(As a reference, we're using the Shankar text "Principles of Quantum Mechanics, 2nd ed.")

Here is my current problem:

(c) Consider the vector space over the complex numbers consisting of all two by two complex matrices M. Show that this is an inner product space with the inner product of two matrices M,M' given by Tr(M†M') where Tr denotes the matrix trace and M† the complex conjugate, transposed M.
(d) With the above inner product, find an orthonormal basis for the space of two by two complex matrices.

I've already solved (c) and am semi-confident about my answer. I'm not really sure if I demonstrated that V is necessarily an inner product space, but I've shown that <M†|M'>=Tr(M†M') which was pretty easy. I think that was the goal of the question, but maybe I'm wrong.

Anyhow, my real struggle is with part (d). I don't really know how to use the inner product to find an orthonormal basis for the space of 2x2 complex matrices. I have a sneaking suspicion that I should be using the Pauli spin matrices and the identity matrix as my linearly independent matrices, and that I need to use the Gram-Schmidt Theorem to form my orthonormal basis. Would anyone agree that finding |M| (where M represents my linearly independent matrices) using the inner product is the way in which they want me to use the inner product to find the orthonormal basis?

[Charlie!]

Graham-schmidt is a good place to go when you don't know what else to do, but it requires starting material.

I suppose you could start with the identity matrix, then find the equation for an arbitrary matrix orthogonal to the identity matrix. Then choose an arbitrary (or not arbitrary) matrix that satisfies this equation, and set up the equation for a matrix that's orthogonal to the first two, etc. Then once you have four orthogonal vectors you can normalize them at your leisure.

[tooyoo]

Concerning (c): Write down the definition of an inner product. (symmetric, linear, positive definite) and verify them one-by-one.

Concerning (d): You should first ask yourself, what the dimension of that vector space is (there is also a difference between complex and real dimension). That way you know, how many basis elements you are looking for.

Formally speaking, Gram-Schmidt is the correct way to do this, yet practically it is not, since you'll probably end up with way to complicated expressions. As a matter of fact, you've already used a good starting point: The Pauli matrices. So I suggest you put the leg work in and simply compute some scalar products of Pauli matrices, which should point you towards a solution.

[Bears!]

Concerning (d): You should first ask yourself, what the dimension of that vector space is (there is also a difference between complex and real dimension). That way you know, how many basis elements you are looking for.

Could you elaborate on this a little? This is sort of where I thought I was getting caught up, because I wasn't certain if it was a four dimensional space given that complex numbers are involved. If they were all matrices consisting of real numbers, I'd say I need four bases. I don't actually know how many I'd need when these matrices consist of complex numbers, though.

[tooyoo]

Sure. First recall that any complex number can be written in terms of two real numbers:
z = u + i v
hence, the real dimension of the complex plane is two while its complex dimension is one. So note that there is a complex and a real dimension. The real dimension is always simply the complex dimension times 2. Concerning basis elements, you can write things either in terms of say "n" basis vector with complex coefficients, or "2n" basis vectors with real coefficients (and factors of "i").

Now, the space you are considering is just the space of matrices without any constraints. You don't need to worry about invertability, unitarity or anything like that. So you should be able to use your argument why the space of real 2x2 matrices is four dimensional and quickly obtain the complex and real dimensions of the space of complex 2x2 matrices.

[Bears!]

So I've basically figured out the solution to the problem, except that I am having trouble demonstrating that Tr(M†M')=Tr(M'†M). I end up getting Tr(M†M')=a*a'+b*b'+c*c'+d*d' and Tr(M'†M)=a'*a+b'*b+c'*c+d'*d, where M and M' are arbitrary matrices, † denotes the complex conjugate, transpose of a matrix, and * represents the complex conjugate of an element of the matrices. These two traces aren't equal, so where did I go wrong?

[tooyoo]

You're calculation is basically correct. You're missing one important fact about the inner product: On real spaces, the inner product is defined to be symmetric. On complex spaces, it's symmetric after complex conjugation. (You call this a Hermitian inner product.)

<v*,w> = (<w^*,v>)*

When you conjugate one of your expressions, you get the other.

Nice work, by the way.

[Bears!]

You're calculation is basically correct. You're missing one important fact about the inner product: On real spaces, the inner product is defined to be symmetric. On complex spaces, it's symmetric after complex conjugation. (You call this a Hermitian inner product.)

<v*,w> = (<w^*,v>)*

When you conjugate one of your expressions, you get the other.

Nice work, by the way.

Yeah, my TA just emailed me about this because I sent him an email. I knew I was missing a conjugation step, since that's the obvious difference between the two sets of coefficients. Well, at least I know what I'm doing now!

Thanks for all the help.

[Bears!]

Alright, it's late and I've all but given up on two homework problems. It's probably going to be brutal reading these since I'm not proficient with jsmath. The first is:

Is it true that a symmetric matrix (that is a matrix S that is equal to its transpose, S = S^T) always remains symmetric under a unitary transformation?

I know it isn't true, but I can't come up with any counter examples. Also, I can't really explain why the explanation for why it isn't true works. Basically, I took the transpose of (U^dagSU) which gave me (U^TS^TU*). I don't really know where to go from there.

The other question is about the Dirac delta function:

Show that the limit $$\delta(x) = \lim_{a->0} \frac{1}{\pi} \frac{a}{a^2+x^2}$$ satisfies the properties of the Dirac delta "function" as defined in Shankar, that is that delta(x) vanishes if x != 0 and that integral from negative infinity to infinity of the Dirac delta equals 1.

I don't really know what I'm suppose to do with the integral. As far as the limit itself goes, I feel it's pretty straightforward. I basically just wrote the limit down and said "it's true," since it's pretty obvious and we're not expected to do epsilon proofs.

[legend]

Q1:
You're heading the right way. Try assuming that U^tSU=U^TSU* and try to get something out of it. It might be easier to get a counterexample for the reformulated problem.
More hints in the spoiler.

Spoiler:

After some algebra you get [UU^T,S]=0. It should be quite easy to disprove that by taking a non-trivial U and than chosing an S so that UU^T doesn't commute with it.

Q2:
I guess you just have to show that \int_{-\infty}^{\infty}\frac{a}{a^2+x^2}dx=\pi for all values of a. Try substitution to get rid of a. After that you get a really classical integral and if you have never seen it I think it's the best to just look it up somewhere. (Alternatively if you know some complex analysis it might be really easy to calculate it that way.)

[Bears!]

Alright, so here's my newest question:

We're given a matrix which is the Hamiltonian:

$$ H = \hbar \omega \left( \begin{array}{ccc}
1 & 0 & 0 \\
0 & 2 & 0 \\
0 & 0 & 2 \end{array} \right) $$

and some matrix (that we're told is an operator, though clearly it actually does nothing):

$$ A = \left( \begin{array}{ccc}
0 & 1 & 0 \\
1 & 0 & 0 \\
0 & 0 & 2 \end{array} \right) $$

I was told to find the eigenvectors and eigenvalues, which I did. I was then told to find the expectation values, which I did. Now I'm asked to evolve a state:

$$ |\psi (0)\rangle = \left ( \begin{array}{ccc} \frac{1}{\sqrt{2}} \\ 0 \\ \frac{1}{\sqrt{2}} \end{array} \right ) $$

I'm not really confident in what I think I should do, but my suspicion is that I simply need to multiply the eigenstates of psi by by$$e^{\frac{-i E_i t}{\hbar}}$$with each corresponding eigenvalue (at least for the Hamiltonian). Is that correct? So then what do I multiply each of the eigenstates with the matrix A? Since this is some random operator, I suspect it doesn't have any time evolution, so what do the components of psi become using the eigenstates of A?

[tooyoo]

Actually, I need to correct that.

You should look up "time evolution operator" in your favorite textbook on QM. Since the Hamiltonian doesn't depend on time explicitly, you'll find something like

U(t,t_0) = exp \frac{-\imath H (t-t_0)}{\hbar}

What you want to calculate is

U(t,0) \vert \psi \rangle

and there are two ways to do this. Since the Hamiltonian is very simple, you can simply perform the exponential and calculate U explicitly. Alternatively you can work in an Eigenbase.

Since you're only asked to evolve a state, you don't need to do anything concerning A. Of course, it's different if you're after the expectation value for A in terms of the state psi. In that case, you'll want something like

\langle \psi(t) \vert A \vert \psi(t) \rangle

which should be easy once you've calculated the time evolution of the state.

In principle, you'll find *all* this in any decent textbook (Sakurai, etc.)

[Bears!]

So is it correct to denote my evolved state like this?:
$$ H |\psi(o)\rangle = e^{-i \omega t} |H = \hbar \omega\rangle + e^{-2 i \omega t} |H = 2\hbar \omega\rangle $$ with those being the components of my eigenbasis corresponding to their respective eigenvalues for H? (I know my questions may seem rudimentary, but this is an introductory quantum class, so we're just getting comfortable with Dirac notation.)

[tooyoo]

Nope. You want to calculate

\vert \psi(t) \rangle = exp \frac{-\imath t H}{\hbar} \vert \psi(0) \rangle

To calculate the exponential, you need to use the standard power-series definition

exp(x) = \sum_{n=0}^\infty \frac{x^n}{n!}

and plug in the matrix (-i t H) for x (ignoring h-bar).

The result is actually very easy. You might want to calculate (-i H)^0, (-i H)^1, (-i H)^2, (-i H)^3 by hand (I'm ignoring the h-bar here), and you should see a pattern emerging. In other words, you'll get a nice big 3x3 matrix

exp \frac{-\imath t H}{\hbar} = \left( 3x3 matrix \right)

that you can multiply directly with your explicit form of the vector psi.

[Bears!]

Alright, can someone very conversationally explain to me the meaning of Poisson brackets? I haven't taken an intermediate mechanics course yet, but we're expected to understand the role of generators and symmetries in my QM class. I have sort of an intuitive picture of what a canonical transformation means - I sort of think of an orthogonal transformation, I guess - but I can't really put any of this into the language of mathematics. I've tried wiki but it's a little higher level for me at this point. Shankar is often times too difficult for me to understand, and this is one of those places where I get really bogged down in his text. Someone care to help?