Find the form of operator ˆx1(t

Quantum Mechanics List of Problems, Section 3 Problem 0 ) A warm up.

Suppose we have a free quantum particle confined in [0, d]. The corresponding Hamiltonian takes the form

H = −¯h² 2m

d² dx² ,

acts in L²(0, d) and the functions from its domain satisfy u(0) = u(d) = 0.

Find available energies and associated eigenvectors.

Problem 1)

Assuming that ˆH = _2m¹ ˆ~p² + ˆV (~r), find a) [ˆp_j, ˆH] ,

b) [ˆx_j, ˆH] , (j = 1, 2, 3).

Problem 2)

Suppose ˆH = _2m¹ ˆ~p² + ˆV (~r). Find the form of operator ˆx₁(t) = ˆx(t) in the Heisenberg picture and position representation for

a) a free particle — ˆV (x) = 0,

b) harmonic oscillator — ˆV (x) = ^{m ω}₂² xˆ². Problem 3)

Check whether the operators ˆx(t₁), ˆx(t₂) expressed in the Heisenberg picture commutate.

Hint: use the result of Problem 2.

Problem 4)

Using the Schr¨odinger equation show the continuity equation

∂ρ

∂t + div ~j = 0, where ρ = |ψ(~r, t)|². Furthermore, show a)

∂

∂t

Z

R³

ρ(~r, t) d³~r = 0, 1

, b) calculate div~j for the stationary states,

c) calculate ~j(ψ_p~(~r) where ψ_~p(~r) is an eigenfuction of operator ˆp.~ Problem 5)

Find the eigenfuctions and eigenvalues of the energy operator for a particle moving in the potential field

V (x) =

( V₀ if 0 ≤ x ≤ a, 0 if x < 0, x > a,

where V₀ > 0 a > 0 (the barrier potential). Find the reflection and transition coefficients.

Problem 6)

a) Find the eigenfunctions and eigenvalues of the energy operator for a par- ticle moving in the potential field

V (x) =

( 0 if 0 ≤ x ≤ a, V₀ if x < 0, x > a, and V₀ > 0 (the well potential).

b) Consider the case V0 → ∞.

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