2

For several years now I have a hobby project in which I consider various QM operators and then calculate how they act on certain wave functions. I evaluate the resulting probability densities and of course the expectation values. I do this in both position representation and momentum representation.

I found a surprising result when I was working with the operator $z^2 p_z^4$. Now there is of course a small issue with the non-commutation between $z$ and $p_z$. I resolved this by considering all $15$ combinations of the 6 operators, and then averaging their outcomes linearly with equal weights of $1/15$. I calculated how the resulting operator acts on the ground state of the Hydrogen atom. The expectation value I found was zero. Quite surprising. Obviously one would expect a positive value since both $z^2$ and $p_z^4$ are positive everywhere.

Here are the $15$ operator sequences, all acting on one wave function $\psi(r)$, and the expectation values I obtained (in units of the Bohr radius $r_{0}$ to the power $-2$): $$1. <zzdddd> = -1 $$ $$2. <zdzddd> = -1/2$$ $$3. <zddzdd> = 0$$ $$4. <zdddzd> = 1/2$$ $$5. <zddddz> = 1$$ $$6. <dzzddd> = 0 $$ $$7. <dzdzdd> = 1/6$$ $$8. <dzddzd> = 1/3$$ $$9. <dzdddz> = 1/2$$ $$10. <ddzzdd> = 1/3$$ $$11. <ddzdzd> = 1/6$$ $$12. <ddzddz> = 0$$ $$13. <dddzzd> = 0$$ $$14. <dddzdz> = -1/2$$ $$15. <ddddzz> = -1$$

How can this result be understood? Does it happen more often in QM that operators have unusual expectation values that go against physical intuition?

M. Wind
  • 761

1 Answers1

3

For arbitrary two operators $A$ and $B$, even if both are positive semi-definite, there's is little to be said about their product. For example, if you take $$ A = \begin{pmatrix}2 & 0 \\ 0 & 0 \end{pmatrix} \quad B = \begin{pmatrix}1 & 1 \\ 1 & 1 \end{pmatrix}, $$ (both are positive semi-definite) and a state $\psi = (0.5, -1)$, then $\langle\psi| (AB + BA)|\psi\rangle = -1$.

So a priori, for non-commuting operators you cannot say much as to what will the expectation value of their product be.

But your expectation value does contain terms that can be proven positive, for example $\langle\psi| z p_z^4 z |\psi\rangle = \Vert p_z^2 z |\psi\rangle \Vert^2 $, which is a norm of a non-zero vector (this will be true for any symmetric string of $z$ and $p_z$). So it must happen that these terms cancel out with other terms in the sum, although I don't see immediately why this would be the case.

Michał Jan
  • 244