A peculiarity in uncertainty principle

Shyan

There is still a point here.
We can't tell that the assumption that A has an eigenstate,causes the paradox because even if P doesn't have an eigenstate,one still can find an operator that does and do the calculation for that operator and its eigenstate and agian arrive at the paradox.So I think sth else should be
wrong.
I've been trying to understand it via reading the thread that Goerge suggested but I can't follow the discussion because I don't know enough math.
Can somebody present the final result of that thread in a simple language?
thanks

Fredrik

I'm not sure about this. I've been saying (incorrectly) in other threads that the box potential is just a way of saying that the Hilbert space is now ##L^2(\text{box})## instead of ##L^2(\mathbb R)##, but now that I think about it, that seems very wrong. Our wavefunctions aren't just zero or undefined outside of the box. They are required to got to zero as we approach the edge of the box. So if the Hilbert space is either of those two spaces I just mentioned, then the energy eigenstates only span a proper subspace of it. We should probably view one of those subspaces as the system's Hilbert space*. In that case, plane waves aren't included.

*) ...or rather, as the system's semi-inner product space, which can be used to define the system's Hilbert space.

Maybe you had periodic boundary conditions in mind, instead of a box potential. I guess that would solve the problem with momentum eigenvectors, but not position eigenvectors.

Anyway, I'm not so sure that non-existence of eigenvectors is really the issue in the 0=1 "proof". The other thread talks mainly about the domains of the operators, so that's probably the real problem here. The commutator [x,p]=i has to be interpreted as in dextercioby's post (in this thread) to make sense.

Shyan

I asked a professor
He said that the problem is,as I understood,that when [itex] [A,B]=cI [/itex] then non of the operators has normalizable eigenvectors.
In fact you can find no A and B for which both [itex] [A,B]=cI [/itex] and [itex] \langle a | a \rangle=1 [/itex],[itex] \langle b | b \rangle =1 [/itex] are true where a and b are eigenvectors of A and B

But I will be more happy if I see a proof of the latter

Dickfore

The problem with x and p is that they have continuous spectra, so their eigenvectors are elements of a rigged Hilbert space. You may get rid of your confusion by the following artifice:

Imagine you impose periodic boundary conditions, so that x and x + L are physically equivalent points. The consequence of this is that momentum acquires only a dicrete set of eigenvalues:
[tex]
k_n = \frac{2\pi n}{L}, \ n = 0, \pm 1, \pm 2, \ldots
[/tex]
and the corresponding eigenfunctions are [itex]\langle x \vert k_n \rangle = L^{-1/2} e_{i k_n x}[/itex], normalized by the condition:
[tex]
\int_{0}^{L} \langle k_m \vert x \rangle \, \langle x \vert k_n \rangle \, dx = \delta_{m,n}
[/tex]
Now, let us expand an arbitrary state ket in the basis of (discrete) momentum eigenvectors. This is nothin more than a Fourier series:
[tex]
\vert \psi \rangle = \sum_{n = -\infty}^{\infty} a_n \, \vert k_n \rangle, \ a_n = \langle k_n \vert \psi \rangle = L^{-1/2} \, \int_{0}^{L} dx e^{-i k_n x} \langle x \vert \psi \rangle
[/tex]
Then, the commutator is:
[tex]
\langle k_m \vert [ x, p ] \vert \psi \rangle \hbar \, \sum_{n} a_n (k_n - k_m) \langle k_m \vert x \vert k_n \rangle
[/tex]
The matrix element of the position operator between discrete momentum eigenstates is:
[tex]
\langle k_m \vert x \vert k_n \rangle = \frac{1}{L} \, \int_{0}^{L} dx x \, e^{-i (k_m - k_n)} = \left\lbrace \begin{array}{lc}
\frac{1}{2} & m = n \\

\frac{i}{k_m - k_n} & m \neq n
\end{array}\right.
[/tex]

Thus, the commuator matrix element is
[tex]
\langle k_m \vert [x, p] \vert \psi \rangle = -i \hbar \sum_{n \neq m} a_n
[/tex]
If you take into account the following identity
[tex]
\sum_{n = -\infty}^{\infty} e^{i \frac{2 \pi n x}{L}} = L \sum_{n = -\infty}^{\infty} \delta(x - n L)
[/tex]
you get the following expression:
[tex]
[x, p] \vert \psi \rangle= i \hbar \left( \vert \psi \rangle - L \sum_{n = -\infty}^{\infty} e^{-\frac{i}{\hbar} n L p} \vert x = 0 \rangle \right)
[/tex]
where the p on the r.h.s. is an operator. Thus, the commutator has changed.

Shyan

I don't understand what you've written in this part.
Looks like some symbols aren't in their place.
Could you write them again and also explain?

And the other point is that,even with this,we can find other operators with the same illness.Can you cure them the same way too?

Thanks

dextercioby

The idea is that they are not cured. It all stems from the x operator being defined the way it is, namely as a mere multiplication by the real variable also denoted by x. This operator has a purely continuous spectrum, its 'eigenvectors' do not fit in the Hilbert space, they are delta Dirac distributions.

K^2

Dickfore, what you've written is consistent with results I got numerically earlier by testing X and P operators in discrete space with periodic boundary conditions. The [X,P] operator ended up having zeroes on diagonal, which immediately tells you it has zero expectation in eigen vectors of X operator. I've separately verified that it has zero expectations for eigen vectors of P as well. So in finite, discrete space, Shyan's proof works, and it is consistent with how we expect commutator to work based on your explanation. So there is nothing strange going on there.

I guess, I just expected the operators to behave very similar in continuous space and discrete space with enough points. It's very interesting to see that there is a qualitative change. I wonder if that says anything about how much we can trust the Lattice calculations.

George Jones

I am working on a long post that gives an example that uses fairly elementary, but somewhat lengthy, calculations to illustrate the problem with domains. If my wife has work (marking) to do tonight, then I might finish the post tonight; if my wife wants to watch a movie, then I won't finish tonight.

Shyan

I don't know much math compared to other ones who have posted here.
I see you guys are really good in this stuff.
But let me give it a try too.
I've tried to figure out what are two general operators which satisfy [itex] [A,B]=cI [/itex] and both are hermitian.
I found the following(Maybe not the most general but general enough)
[itex]
A=a \frac{d}{d \alpha}+f(\alpha) \\
B=b \alpha
[/itex]
Now consider:
[itex]
A \psi(\alpha)=\lambda \psi(\alpha) \Rightarrow \psi(\alpha)=A e^{- \int \frac{f(\alpha)-\lambda}{a} d\alpha}
[/itex]
You see that the above function isn't normalizable.
We also have:
[itex]
B \psi(\alpha)=\lambda \psi(\alpha) \Rightarrow b \alpha \psi(\alpha)=\lambda \psi(\alpha) [/itex]
Which can't be true unless [itex] \alpha [/itex] is a constant which can't be.
So when [itex] [A,B]=cI [/itex] and both A and B are hermitian,One of them will not have normalizable eigenstates and the other doesn't have eigenstates at all and the assumption of their existence is the wrong part of the proof.
I will be happy to hear corrections.
Thanks all

dextercioby

Take [itex] f(\alpha) = e^{-\alpha^2} [/itex] with [itex] \alpha\in\mathbb{R} [/itex]

Shyan

I guess you meant:
[itex]
f(\alpha)=2ca \alpha+\lambda \Rightarrow \psi(\alpha)=A e^{-c \alpha^2}
[/itex]
Which is normalizable.
Well,I think I'll wait for the answer!

dextercioby

Yes, I thought of the convergent exponential in the shape of a gaussian. So you can find potentials for which the wavefunction is a well-behaved mathematical object.

Shyan

So we have concluded that the problem is NOT the assumption of the existence of A's and B's normalizable eigenstates?

Shyan

In Quantum mechanics:Concepts and applications by Nouredine Zettili (2nd edition),one can find the following:
[itex]
| \psi \rangle = \int d^3 r | \vec{r} \rangle \langle \vec{r} | \psi \rangle = \int d^3 r \psi(\vec{r}) | \vec{r} \rangle
[/itex]
With [itex] \langle \vec{r} | \psi \rangle = \psi(\vec{r}) [/itex]
I tried to write the paradox with this convention:
[itex]

1=\langle a | a \rangle=\frac{1}{c} \langle a | cI | a \rangle=\frac{1}{c} [ \int d^3 r a^*(\vec{r}) (cI) \int d^3 r a(\vec{r}) ] | \vec{r} \rangle=\frac{1}{c} [ \int d^3 r a^*(\vec{r}) [A,B] \int d^3 r a(\vec{r}) ] | \vec{r} \rangle=\frac{1}{c} [ \int d^3 r a^*(\vec{r}) AB \int d^3 r a(\vec{r})-\int d^3 r a^*(\vec{r}) BA \int d^3 r a(\vec{r}) ] | \vec{r} \rangle

[/itex]
I'm not sure how to continue but I think it may help.So I give it to more capable hands.

Shyan

Yeah,again me.
This time i asked a professor of pure mathematics.
He said the problem is that [itex] [A,B] | \psi \rangle =c |\psi \rangle [/itex] doesn't mean [itex] [A,B]=cI [/itex]
I think the reason is that the domain of I is the whole space but the domain of [A,B] is more restricted so the two operators only have the same prescription but different domains so they're not equal.

Dickfore

I think the simplest way to put it is that [itex]\langle a \vert a \rangle[/itex] and [itex]\langle a \vert B \vert a \rangle[/itex], where [itex]\vert a \rangle[/itex] is an eigenvector of A formally do not exist (they diverge, are infinite)!
Calculate the M.E. of this commutation relation between two different eigenstates:
[tex]
\langle a \vert [A, B] \vert a' \rangle = (a - a') \, \langle a \vert B \vert a' \rangle = c \langle a \vert a' \rangle
[/tex]
When [itex]a' \neq a[/itex], the r.h.s. is zero, so we conclude [itex]\langle a \vert B \vert a' \rangle = 0, \ a' \neq a[/itex]. But, if we take the limit [itex]a' \rightarrow a[/itex], the l.h.s. is an indeterminate form of the type [itex]0 \cdot \infty[/itex], whereas the r.h.s. is [itex]c \, \delta(a - a')[/itex], according to the orthonormalization condition for continuous spectra.

This was the motivation behind making the eigenvalues of at least one of the operators A or B, discrete (in my case, the momentum operator p), because then you may normalize everything to a finite norm.

Shyan

Ok,but can you prove [itex] \langle a | B | a \rangle=\infty [/itex] and [itex] \langle a | a \rangle=\infty [/itex] for all A,B and [itex] |a \rangle [/itex] satisfying the conditions?