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Quantum Physics I (8.04) Spring 2016
Assignment 3
MIT Physics Department Due Thu. February 25, 2016 February 18, 2016 5:00pm
Announcements
- Recommended Reading: Griffiths, sections 1.1, 1.2, 1.4 and 1.5.
Problem Set 3
- Exercises with commutators [10 points] Let A, B, and C be linear operators.
(a) Show that [A, BC] = [A, B]C + B[A, C]. (b) Show that [AB, C] = A[B, C] + [A, C]B. (c) Show that [A, [B, C]] + [B, [C, A]] + [C, [A, B]] = 0. (d) Calculate [xˆn, pˆ] and [x, pˆn] for n an arbitrary integer greater than zero. (e) Calculate [xˆpˆ, xˆ^2 ] and [xˆpˆ, pˆ^2 ].
- Simple tests of the stationary phase approximation [10 points] In here we consider integrals of the form
Ψ(x) =
dk Φ(k)eikx^ , −∞
where Φ(k) is a function that is sharply localized around k = k 0. In each of the following cases use the stationary phase argument to predict the location of the peak of |Ψ(x)|. Then compute the integral exactly to find Ψ(x), |Ψ(x)|, and to confirm your prediction.
(a) Φ(k) = e−L^2 (k−k^0 )^2 , where L is a constant with units of length. (b) Φ(k) = e−L^2 (k−k^0 )^2 e−ikx^0 , where x 0 and L are constants with units of length.
Useful integral: Valid for complex constants a and b, with real part of a positive: ∫ (^) ∞ e−ax
(^2) +bx dx = −∞
π a
exp
( (^) b 2 , 4 a
when Re(a) > 0.
Physics 8.04, Quantum Physics 1, Spring 2016 2
- Galilean invariance of the free Schrodinger equation. [15 points] Show that the free-particle one-dimensional Schro¨dinger equation for the wavefunc- tion Ψ(x, t): ∂Ψ iℏ ∂t
ℏ^2
2 m
∂^2 Ψ
∂x^2 is invariant under Galilean transformations
x′^ = x − vt , t′^ = t.
By this we mean that there is a Ψ′(x′, t′) of the form
Ψ′(x′, t′) = f (x, t) Ψ(x, t) ,
where the function f (x, t) involves x, t, ℏ, m and v, and such that Ψ′^ satisfies the corresponding Schro¨dinger equation in primed variables.
∂Ψ′ iℏ ∂t′^
ℏ^2
2 m
∂^2 Ψ′
∂x′^2 (a) Find the function f (x, t). [Hint: Note that the function f (x, t) cannot depend on any observable of Ψ; it is a universal function that is used to transform any Ψ. Thus if Ψ is a (single) plane wave, f cannot depend on its momentum or its energy.] (b) Demonstrate that the plane wave solution
Ψ(x, t) = A ei(kx−ωt)
transforms as expected. In other words, give Ψ′^ and show that it represents, in the primed reference frame, a particle with the expected momentum and energy.
- Re-do current conservation in 3D [10 points] In class we derived the expression for the one-dimensional probability current J(x, t) starting from ρ(x, t) = |Ψ(x, t)|^2 and using the one-dimensional Schro¨dinger equation to write ∂ρ ∂t
∂J
∂x Repeat the same steps starting from
ρ(x, t) = ∣∣Ψ(x, t)∣^2 ,
and using the three-dimensional Schro¨dinger equat
ion to derive the form of the prob- ability current J(x, t) that should appear in the conservation equation
∂ρ
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