## Physics Friday 13

In quantum mechanics, observables correspond to Hermitian operators on the quantum state . This state is a vector in a complex Hilbert space; we have the inner product between two vectors, which is  in bra-ket notation.

Now, for a Hermitian operator  corresponding to some observable quantity, the expectation value of that observable for a quantum state  is .

Now, remember that for a Hilbert space, as for all inner product spaces, we have the norm . We also note that for any two Hermitian operators  and  and any state vector , we have
,
where the bar denotes the complex conjugate. These arise because the operators are Hermitian.

For two operators A and B, the Cauchy Schwarz inequality tells us


Now,





,
where  is the commutator of A and B.

However, we note


,
and similarly for B. Thus, our above inequality can be written:


Now, we recall in probability theory that the variance of a random variable X is
, and the standard deviation is the square root of the variance. Now, for two operators A and B, let us define  and .
We see that as  and  are numbers,





Suppose we define for A a quantity ΔA analogous to the standard deviation for a random variable:
. Let us also analogously define ΔB. Then we see that
.
Using the inequality we have derived, we see that this means
, or, since ,
.

This inequality is, in fact, the general form of the Heisenberg Uncertainty Principle; it tells us the uncertainty relation between any two observables whose corresponding operators do not commute. For example, given the (one-dimensional) position operator X and corresponding momentum operator P, we have , so  for a valid (normalized) state vector, giving us the usual form of the Uncertainty Principle:
.