1. Suppose Q and P are Hermitian operators on a Hilbert space of finite dimension dim H = d. Show that these operators cannot possibly have the canonical commutation relation 2. The remarkable Wigner...

1. Suppose Q and P are Hermitian operators on a Hilbert space of finite dimension dim H = d. Show that these operators cannot possibly have the canonical commutation relation

2. The remarkable Wigner function was introduced by Eugene Wigner in 1932. It is a useful alternative representation of quantum states. Suppose a particle is moving in 1-D and is described by a wave function ψ(x). Then the Wigner function is

The Wigner function has many of the characteristics of a joint probability distribution over x and p. Prove the following properties of P(x, p):

If we integrate over either x or p, we obtain the correct quantum probability distributions over p and x. However, the Wigner function cannot be properly regarded as a joint probability distribution. To prove this, construct an example in which P(0, 0) <>