19.5 Homework
- Att= 0, a 1D harmonic oscillator is in a linear combination of the energy eigenstates
ψ=
√
2
5
u 3 +i
√
3
5
u 4
Find the expected value ofpas a function of time using operator methods.
- Evaluate the “uncertainty” inxfor the 1D HO ground state
√
〈u 0 |(x− ̄x)^2 |u 0 〉where ̄x=
〈u 0 |x|u 0 〉. Similarly, evaluate the uncertainty inpfor the ground state. What is the product
∆p∆x? Now do the same for the first excited state. What is the product ∆p∆xfor this state?
- An operator is Unitary ifUU†=U†U= 1. Prove that a unitary operator preserves inner
products, that is〈Uψ|Uφ〉=〈ψ|φ〉. Show that if the states|ui〉are orthonormal, that the
statesU|ui〉are also orthonormal. Show that if the states|ui〉form a complete set, then the
statesU|ui〉also form a complete set.
- Show at if an operatorHis hermitian, then the operatoreiHis unitary.
- Calculate〈ui|x|uj〉and〈ui|p|uj〉.
- Calculate〈ui|xp|uj〉by direct calculation. Now calculate the same thing using
∑
k
〈ui|x|uk〉〈uk|p|uj〉.
- Ifh(A†) is a polynomial in the operatorA†, show thatAh(A†)u 0 =dh(A
†)
dA† u^0. As a result of
this, note that since any energy eigenstate can be written as a series of raising operators times
the ground state, we can representAbydAd†.
- Att= 0 a particle is in the one dimensional Harmonic Oscillator stateψ(t= 0) =√^12 (u 0 +u 1 ).
- Compute the expected value ofxas a function of time usingAandA†in the Schro ̈dinger
picture.
- Now do the same in the Heisenberg picture.
