Time-Dependence of Expectation Values

Question 1

What is the strategy to find d/dt()?

Answer 1

• Expand d/dt in partial derivatives: dQ(hat)/dt, dΨ*/dt, dΨ/dt
• Sub dΨ/dt = H(hat)Ψ/iћ, dΨ/dt = H(hat)Ψ*/-iћ
• Use Hermitian nature to get d/dt = + i/ћ *

Question 2

When does time dependence arise?

Answer 2

Due to both time variation of the operator and the commutator of the operator with the Hamiltonian.

Question 3

What is d/dt of the position operator x(hat) in 1D?

Answer 3

d/dt = i/2mћ *

Question 4

If we consider [A, BC] = ABC - BCA, how do we compute this?

Answer 4

-Insert 2 terms -BAC and +BAC (as they add to zero) in the middle = (AB-BA)C + B(AC-CA) = [A,B]C + B[A,C]

Question 5

What do we find equals?

Answer 5

= -2iћ *<p></p>

Question 6

What is the final equation for d/dt?

Answer 6

d/dt = <p>/m</p>

Question 7

What is the equation for p(x)(hat) in terms of ћ?

Answer 7

p(x)(hat) = -iћ d/dx, so dp(x)(hat)/dt = 0

Question 8

What does [A+B, C] equal?

Answer 8

=[A+B, C] = [A, C] + [B, C]

Question 9

What is the equation for d/dt <p>?</p>

Answer 9

d/dt(<p>) = i/ћ *</p>

Question 10

How do we work out the value of [V(hat), p(x)(hat)]?

Answer 10

Consider [V(hat), p(x)(hat)]Ψ = V(hat)p(x)(hat)Ψ - p(x)(hat)V(hat)*Ψ, then sub in equation for p(x)(hat) = iћ d/dx and V(hat) = V, then rearrange

Question 11

What is the final equation for d/dt <p>? after rearranging?</p>

Answer 11

d/dt <p> = - : rate of change of momentum = force (Newtons 2nd)</p>

Question 12

What do we learn from a constant potential dV/dx = 0?

Answer 12

dp/dt = 0, so momentum is conserved.

Question 13

What do we get if we combine d/dt = <p>/m, and d/dt </p><p> = -?</p>

Answer 13

Get m*d^2/dt^2 = F = - : F = ma