Physics Problems & Solutions: Momentum Operator

Showing posts with label Momentum Operator. Show all posts

Quantum Mechanics - Momentum Operator

The wave function of a particle is e^i(kx−ωt) where x is distance, t is time, and k and are ω positive real numbers. The x–component of the momentum of the particle is

A. 0
B. ħω
C. ħk
D. ħω/c
E. ħk/ω

(GR9277 #01)

Solution:

Momentum operator, p = −iħ∇

pψ = −iħ ∂ψ/∂x

= −iħ ∂e^i(kx−ωt)/∂x

= ħk e^i(kx−ωt)

= ħkψ

Answer: C

Quantum Mechanics - Infinite Potential Well

Questions 51-53

A particle of mass m is confined to an infinitely deep square-well potential:

V(x) = ∞, x ≤ 0, x ≥ a
V(x) = 0, 0

a

The normalized eigenfunction, labeled by the quantum number n, are

$\psi_n=\sqrt{\frac{2}{a}} \sin \frac{n\pi x}{a}$

For any state n, the expectation value of the momentum of the particle is

A. 0

B. $\frac{\hbar n \pi}{a}$

C. $\frac{2\hbar n \pi}{a}$

D. $\frac{\hbar n \pi}{a} \left ( \cos n\pi-1 \right )$

E. $\frac{-i\hbar n \pi}{a} \left ( \cos n\pi-1 \right )$

(GR9277 #51)

Solution:

Infinitely deep square-well potential
→ there is zero probability for particle to be outside the well
→ 〈 $\hat p$ 〉= 0

If 〈 $\hat p$ 〉 ≠ 0 the particle would tend to go to the right or left and leave the well, which is impossible for infinitely deep square-well potential.

Answer: A

Alternative Answer #1:

Eigen function, $\hat p$ ψ_n = λψ_n
The eigenvalue, λ is associated with expectation value:〈Â〉= 〈ψ | Â | ψ〉

Since $\hat p$ is imaginer and ψ_n is real → the eigenvalue, λ is imaginer = not real = not observable
→ 〈 $\hat p$ 〉= 0

Alternative Answer #2:

since sine and cosine are orthogonal the whole period.

Quantum Mechanics - Schrodinger Equation

The Hamiltonian operator in the Schrodinger equation can be formed from the classical Hamiltonian by substituting

A. Wavelength and frequency for momentum and energy
B. A differential operator for momentum
C. Transition probability for potential energy
D. Sums over discrete eigenvalues for integrals over continuous variables
E. Gaussian distributions of observables for exact values

(GR8677 #49)

Solution:

Schrodinger Equation: Hψ(x) = Eψ(x)

Hamiltonian: $H=\frac{\hat p^2}{2m}+V(x)$

Momentum operator : $\hat p=-i\hbar \frac{\partial }{\partial x}$

Answer: B

Quantum Mechanics - Eigenfunction

Which of the following is an eigenfunction of the linear momentum operator −iħ ∂/∂x with a positive eigenvalue ħk; i.e., an eigenfunction that describes a particle that is moving in free space in the direction of positive x with a precise value of linear momentum?

A. cos kx
B. sin kx
C. e^−ikx
D. e^ikx
E. e^−kx

(GR8677 #57)

Solution:

Eigenequation:

$\begin{aligned} \hat{p}\psi&=a\psi \\-i\hbar\frac{\partial \psi}{dx}&=\hbar k\psi \\\frac{\partial \psi}{dx}&=ik\psi \end{aligned}$

(A) FALSE
cos kx → ∂(cos kx)/∂x = sin kx

(B) FALSE
sin kx → ∂(sin kx)/∂x = cos kx

(C) FALSE
e^−ikx → ∂(e^−ikx)/∂x = −ik e^−ikx

(D) TRUE
e^ikx → ∂(e^ikx)/∂x = ik e^ikx

(E) FALSE
e^−kx→ ∂(e^kx)/∂x = k e^kx

Answer: D