## Harmonic Oscillator position and momentum Hamiltonian operators

Posted by peeterjoot on December 18, 2010

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# Motivation.

Hamiltonian problem from Chapter 9 of [1].

## Problem 1.

### Statement.

Assume and to be Heisenberg operators with and . For a Hamiltonian corresponding to the harmonic oscillator show that

### Solution.

Recall that the Hamiltonian operators were defined by factoring out the time evolution from a set of states

So one way to complete the task is to compute these exponential sandwiches. Recall from the appendix of chapter 10, that we have

Perhaps there is also some smarter way to do this, but lets first try the obvious way.

Let’s summarize the variables we will work with

The operator in the exponential sandwich is

Note that the constant factor will commute with all operators, which reduces the computation required

For , or , we’ll want some intermediate results

and

Using these we can evaluate the commutators for the position and momentum operators. For position we have

Since , we have

For the momentum operator we have

So we have

The expansion of the exponential series of nested commutators can now be written down by inspection and we get

Collection of terms gives us the desired answer

# References

[1] BR Desai. *Quantum mechanics with basic field theory*. Cambridge University Press, 2009.

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