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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.

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