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Harmonic oscillator problems

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The Hamiltonian of the harmonic oscillator is:

H = P^2/(2m) + ½m omega^2 X^2

You can rewrite this as:

H = h-bar omega [a^(+) a^(-) + ½] (1)

Here:

a^(-) = Squareroot[m omega/(2 hbar)] [X + i/(m omega) P] (2)

and

a^(+) = Squareroot[m omega/(2 hbar)] [X - i/(m omega) P] (3)

The operators a^(-) and a^(+) are each other Hermitian conjugates. Using (2) and (3) and the fact that [X,P] = i h-bar you find that the commutator is:

[a^(-) ,a^(+)] = 1 (4)

You can now easily derive that the eigenvalues of the operator N = a^(+)a^(-) are integers greater than or equal to zero. Using (4) you easily derive that:

[N, a^(-)] = -a^(-) (5)

[N, a^(+)] = a^(+) (6)

Let's denote by |y> the eigenstate of N with eigenvalue x. Then using (5) let's find out how the operator N acts on the state a^(-)|y>:

Na^(-) |y> - a^(-) N|y> = -a^(-)|y> --->

Na^(-) |y> - xa^(-) |y> = -a^(-)|y> --->

Na^(-) |y> = (y - 1)a^(-)|y>

So, if there is an eigenstate with eigenvalue y, denoted as |y>, then a^(-)|y> is an eigenstate with eigenvalue y - 1. This is provided we can properly normalize this new state. Similarly you find using (6) that a^(+)|y> is an eigenstate with eigenvalue y + 1. We must properly normalize these states, so let's find out what the norm is. The inner product of a^(-)|y> with itself is:

<y|a^(+) ...

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