## Tag Archives: quantum mechanics

George Jones put up a neat little puzzle on physicsforums. Following up on this taught me quite a bit about some of the mathematical subtleties of QM that physicists tend to gloss over.

“Suppose A is an observable, i.e., a self-adjoint operator, with real eigenvalue a and normalized eigenket $\left| a \right>$. In other words,

$A \left| a \right> = a \left| a \right>, \hspace{.5 in} \left< a | a \right> = 1.$

Suppose further that A and B are canonically conjugate observables, so

$\left[ A , B \right] = i \hbar I$,

where I is the identity operator. Compute, with respect to $\left| a \right>$, the matrix elements of this equation divided by $i \hbar$:

$\frac{1}{i \hbar} \left< a | \left[ A , B \right] | a \right>)= \left< a | I | a \right>$
$\frac{1}{i \hbar} \left( \left< a | AB | a \right> - \left \right) = \left< a | I | a \right>$

In the first term, let A act on the bra; in the second, let A act on the ket:

$\frac{1}{i \hbar} \left( a \left< a | B | a \right> - a \left \right)= $.

Thus,

0 = 1.

This is my favourite “proof” of the well-known equation 0 = 1.

What gives?”

When you’re done racking your brains, you can take a look here, here and on the arxiv for solutions to this and more such QM puzzlers. The last link is to a paper by F. Gieres called ‘Mathematical surprises and Dirac’s formalism in Quantum Mechanics’, which I found pretty fascinating. Apparently, when we’re taught as undergrads that state vectors in QM live in a Hilbert space, that’s one of those white lies that teachers tell.

This is a continuation of the previous post on Green’s Functions.

When you have a hammer, every problem looks like a nail.

Take the Schrodinger equation: $\hat{H} \psi(\vec{x},t) = i h \frac{\partial\psi(\vec{x},t)}{\partial t}$

Writing it as $(-\frac{h^2}{2 m}\nabla^2 - i h \frac{\partial}{\partial t}) \psi(\vec{x},t)=-V(\vec{x},t)\psi(\vec{x},t)$, we can try and come up with a green’s function for it.

What we’d like to be able to do is start with the wavefunction and potential at some point in time, and have it tell us the wavefunction for all times.

The standard QM thing to do would be to write the wavefunction as the sum of eigenstates of the Hamiltonian, them time evolve each of them. But instead, let’s see how far Mr. Green can take us.