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Phys 500, Quantum Mechanics Homework 2 - UBC …raussen/Phys500/Assignments/HW2… · ·...
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Phys 500, Quantum Mechanics Homework 2 Posted: Wed, October 1, 2014 • Due: Fri, October 10, 2014, 1PM. Problem 1 (5 points). Show that the Gaussian wave packet |ψi with ψ(x) := hx|ψi = (2πd 2 ) -1/4 exp - x 2 4d 2 (1) saturates the Heisenberg uncertainty relation, i.e., p h(Δx) 2 i p h(ΔP ) 2 i = ~ 2 . You may assume the representation of the momentum operator in the position basis ˆ P = -i~ Z dx|xi d dx hx|. (2) Problem 2 (5 points): Consider a particle subject to a one-dimensional simple harmonic oscillator potential. Suppose at t = 0 the state vector is given by exp -i a ˆ P ~ ! |0i, where ˆ P is the momentum operator, a a distance and |0i the lowest number-eigenstate, i.e. ˆ N |0i = 0. Evaluate h ˆ X i, for t ≥ 0, using the Heisenberg picture. Problem 3. (5 points) Consider two Hermitian operators ˆ X , ˆ P acting on a Hilbert space of dimension d. Can the commutation relation [ ˆ X, ˆ P ]= i~ be realized for finite d? Justify your answer. Problem 4 (5 points): Starting from the position-momentum commutation relation [x i ,p j ]= i~δ ij I , prove the relations [x i ,F (p)] = i~ ∂F ∂p i , [p i ,G(x)] = -i~ ∂G ∂x i we used in the derivation of Ehrenfest’s theorem. Problem 5 (5 points): Consider a spin-1/2 particle evolving in a magnetic field under the Hamiltonian ˆ H = ω ˆ S z . Compute the expectation value h ˆ S x (t)i, given the initial conditions h ˆ S x (0)i, h ˆ S y (0)i. Starting from ˆ S x (t) = exp i ˆ Ht ~ ! ˆ S x (0) exp -i ˆ Ht ~ ! , derive the result in two ways, namely 1
Transcript of Phys 500, Quantum Mechanics Homework 2 - UBC …raussen/Phys500/Assignments/HW2… · ·...
Phys 500, Quantum Mechanics
Homework 2
Posted: Wed, October 1, 2014 • Due: Fri, October 10, 2014, 1PM.
Problem 1 (5 points). Show that the Gaussian wave packet |ψ〉 with
ψ(x) := 〈x|ψ〉 = (2πd2)−1/4 exp
(− x2
4d2
)(1)
saturates the Heisenberg uncertainty relation, i.e.,√〈(∆x)2〉
√〈(∆P )2〉 = ~
2. You may
assume the representation of the momentum operator in the position basis
P = −i~∫dx|x〉 d
dx〈x|. (2)
Problem 2 (5 points): Consider a particle subject to a one-dimensional simple harmonicoscillator potential. Suppose at t = 0 the state vector is given by
exp
(−iaP
~
)|0〉,
where P is the momentum operator, a a distance and |0〉 the lowest number-eigenstate, i.e.N |0〉 = 0. Evaluate 〈X〉, for t ≥ 0, using the Heisenberg picture.
Problem 3. (5 points) Consider two Hermitian operators X, P acting on a Hilbert spaceof dimension d. Can the commutation relation [X, P ] = i~ be realized for finite d? Justifyyour answer.
Problem 4 (5 points): Starting from the position-momentum commutation relation [xi, pj] =i~δijI, prove the relations
[xi, F (p)] = i~∂F
∂pi, [pi, G(x)] = −i~∂G
∂xi
we used in the derivation of Ehrenfest’s theorem.Problem 5 (5 points): Consider a spin-1/2 particle evolving in a magnetic field under theHamiltonian H = ωSz. Compute the expectation value 〈Sx(t)〉, given the initial conditions〈Sx(0)〉, 〈Sy(0)〉. Starting from
Sx(t) = exp
(iHt
~
)Sx(0) exp
(−iHt
~
),
derive the result in two ways, namely
1
(a) Use the Baker-Hausdorff formula1, and the spin commutation relations [Si, Sj] =
i~εijkSk, for i, j, k ∈ {x, y, z}.
(b) Find a 2× 2 matrix representation of exp(−i Ht
~
)and Sx. You may use the relation
σiσk = δi,kI + iεiklσl among the Pauli matrices 2.
Total: 25 points.
1The Baker-Hausdorff formula is
exp(iλG)A exp(−iλG) = A+ iλ[G,A] +(iλ)2
2![G, [G,A]] + ..+
(iλ)n
n![G, [G, [G, ..[G,A]..]]] + ..,
where λ is a real parameter, G a Hermitian operator and A a linear operator. See Sakurai (1994), p. 96 orSakurai and Napolitano (2010), p. 95.
2The Pauli matrices σx, σy and σz are 2× 2-matrices defined as follows
σx =
(0 11 0
), σy =
(0 −ii 0
), σz =
(1 00 −1
).
2
