- A 1D harmonic oscillator is in a linear combination of the energy eigenstates

Find the expected value of .

**Answer**

- Assuming represents the 1D harmonic oscillator energy eigenstate,
calculate
.

**Answer**

- Evaluate the ``uncertainty'' in for the 1D HO ground state
where
.
Similarly, evaluate the uncertainty in for the ground state.
What is the product
?

**Answer**

Its easy to see that either from the integral or using operators.
I'll use operators to compute the rest.

- Use the commutator relation between and to derive .
Now show that is the
**lowering operator for the harmonic oscillator** energy.
- At , a
**one dimensional harmonic oscillator** is in the state
.
Calculate the expected value of as a function of time.
- At , a harmonic oscillator is in a linear combination of the and states.

Find
and
as a function of time.
- A 1D harmonic oscillator is in a linear combination of the energy eigenstates

Find
.

Jim Branson
2013-04-22