130_notes.dvi

state of the particle and on the potential.

Answer

d〈A〉

dt

=

1

i ̄h

〈[A,H]〉

d〈x〉

dt

=

1

i ̄h

〈[

x,

p^2

2 m

]〉

=

1

i ̄h

〈

p

m

(

− ̄h

i

)〉

=

〈p〉

m

d〈p〉

dt

=

1

i ̄h

〈[p,V(x)]〉=

1

i ̄h

̄h

i

〈[

d

dx

,V(x)]〉

= −

〈

dV

dx

〉

3. Compute the commutators [A†,An] and [A,eiHt] for the 1D harmonic oscillator.
   Answer

[A†,An] = n[A†,A]An−^1 =−nAn−^1

[A,eiHt] = [A,

∑∞

n=0

(it)nHn

n!

] =

∑∞

n=0

(it)n[A,Hn]

n!

=

∑∞

n=0

n(it)n[A,H]Hn−^1

n!

=it

∑∞

n=1

(it)n−^1 ̄hωAHn−^1

(n−1)!

it

∑∞

n=1

(it)n−^1 ̄hωAHn−^1

(n−1)!

= it ̄hωA

∑∞

n=1

(it)n−^1 Hn−^1

(n−1)!

= it ̄hωA

∑∞

n=0

(it)nHn

(n)!

=it ̄hωAeiHt

4.*Assume that the states|ui>are the eigenstates of the Hamiltonian with eigenvaluesEi,
(H|ui>=Ei|ui>).

a) Prove that< ui|[H,A]|ui>= 0 for an arbitrary linear operatorA.

b) For a particle of massmmoving in 1-dimension, the Hamiltonian is given byH=p

2

2 m+

V(x). Compute the commutator [H,X] whereXis the position operator.

c) Compute< ui|P|ui>the mean momentum in the state|ui>.

5.*Att= 0, a particle of massmis in the Harmonic Oscillator stateψ(t= 0) =√^12 (u 0 +u 1 ).
Use the Heisenberg picture to find the expected value ofxas a function of time.