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Compute the uncertainty product for a harmonic oscillator

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Notation: If A is an operator then I'll denote the Hermitian conjugate of A by C(A)

The annihilation operator in terms of p and x is:

a = 1/sqrt(2m hbar omega) [m omega x + i p]

And the creation operator is its Hermitian conjugate:

C(a) = 1/sqrt(2m hbar omega) [m omega x - i p]

It is convenient to have the expressions for x and p in terms of a and C(a). Solving the above equations for x and p gives:

x = Squareroot[h-bar/(2 m omega)] [a + C(a)]

p = i Squareroot[m omega h-bar/2] [C(a) - a]

The Hamiltonian is:

H = [C(a) a + 1/2] hbar omega

a and C(a) satisfy the commutation relation:

[a, C(a)] = 1 (1)

We also have:

[H, a] = [[C(a) a + 1/2] hbar omega, a] = hbar omega [C(a) a, a] = hbar omega [C(a)a a - a C(a) a ]

From (1) it follows that a C(a) = 1 + C(a) a. The second term in the above formula can thus be written as:

a C(a) a = [1 + C(a) a] a = a + C(a) a a

The expression for the commutator thus simplifies to:

[H, a] = -hbar omega a (2)

We can compute the eigenvectors and eigenvalues of H as follows. We put C(a) a = N. Then since H = hbar omega(N+1/2)
we can just as well compute the eigenstates of N. These are then also the eigenstates of H. If |n> is a normalized eigenstate of N with eigenvalue n, then N|n> = n|n> and thus H|n> = (n+ 1/2)hbar omega|n>, so the corresponding eigenvalue of H is
(n + 1/2)hbar omega. Note that we are not assuming that n is an integer.

Consider the action of N on the state a|n>:

N a|n> = C(a) a a|n>

Eq. (1) says that

a C(a) - C(a) a = 1 ------>

C(a) a = a C(a) - 1

and we see that

N a|n> = C(a) a a|n> = [a C(a) - 1]a |n> = a C(a) a|n> - a|n> = a N|n> - a|n> = n a |n> - a |n> = (n-1)|n> (3)

So, it looks like a|n> is an eigenstate with eigenvalue n-1. Let's compute the norm of this state. The absolute value squared of the norm is:

<n|C(a) a|n> = <n|N|n> = <n|n|n> = n <n|n> = n. (4)

The absolute value squared of any ...

Solution Summary

A detailed solution is derived from first principles.