Physics Problems & Solutions: Eigenstates

Showing posts with label Eigenstates. Show all posts

Quantum Mechanics - Infinite Potential Well

A measurement of energy E will always satisfy which of the following relationships?

A. E ≤ π²ħ²/8ma²

B. E ≥ π²ħ²/2ma²
C. E = π²ħ²/8ma²
D. E = n²π²ħ²/8ma²
E. E = π²ħ²/2ma²

(GR9277 #53)

Solution:

Wave function, $\psi_n=\sqrt{\frac{2}{a}} \sin \frac{n\pi x}{a}$
Time Independent Schrodinger Equation: $-\frac{\hbar^{2}}{2m}\psi''+ V\psi=E\psi$

Inside the well, V(x) = 0, and

$\\\psi=A\sin kx \\\psi'=Ak\cos kx \\\psi''=-Ak^2\sin kx$

$\\-\frac{\hbar^{2}}{2m}(-Ak^2 \sin kx)=EA\sin kx\rightarrow E=\frac{\hbar^{2}k^2}{2m}$

$\\k=\frac{n\pi}{a}\rightarrow E=\frac{n^2\pi^2 \hbar^{2}}{2ma^2} \\\\n=1,2,3, ...\rightarrow E\geqslant \frac{\pi^2 \hbar^{2}}{2ma^2}$

Answer: B

Quantum Mechanics - Normalization of Wavefunction

Eigenfunction for a rigid dumbbell rotating about its center have a φ dependence of the form ψ(φ) = Ae^imφ, where m is a quantum number and A is a constant. Which of the following values of A will properly normalize the eigenfunction?

A. √(2π)
B. 2π
C. (2π)²
D. 1/√(2π)
E. 1/(2π)

(GR8677 #28)

Solution:
Normalized Wavefunction: $\int |\psi(x)|^2dx=\int \psi(x)\psi^*(x)dx=1$
$\\\psi(\phi)=Ae^{im\phi} \\\psi^*(\phi)=Ae^{-im\phi}$

$\\\int_{0}^{2\pi} |\psi(\phi)|^2d\phi=\int_{0}^{2\pi} |A|^2d\phi=1 \\|A|^2\int_{0}^{2\pi}d\phi=|A|^2 2\pi=1\rightarrow A = \frac{1}{\sqrt{2\pi}}$

Answer: D

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

Quantum Mechanics - Expectation Value

The state $\psi=\frac{1}{\sqrt 6}\psi_{-1} +\frac{1}{\sqrt 2}\psi_1 +\frac{1}{\sqrt 3}\psi_2$ is linear combination of three orthonormal eigenstates of the operator $\hat{O}$ corresponding to eigenvalues -1,1 and 2. What is the expectation value of $\hat{O}$ for this state?

A. 2/3
B. √(7/6)
C. 1
D. 4/3
E. (√3+2√2-1)/√6

(GR0177 #29)

Solution:

The expectation value:

$\begin{aligned} \langle \hat{O}\rangle &=\langle\psi|\hat{O}|\psi\rangle \\&=O_1\left(\frac{1}{\sqrt{6}}\right )^2\langle \psi_{-1}|\psi_{-1}\rangle +O_2\left(\frac{1}{\sqrt 2}\right )^2\langle\psi_1|\psi_1\rangle +O_3\left( \frac{1}{\sqrt{3}}\right )^2\langle\psi_2|\psi_2\rangle \\&= (-1)\frac{1}{6} +(1)\frac{1}{2}+(2)\frac{1}{3}= 1\end{aligned}$

Answer: C

Quantum Mechanics - Particle in a Box

The energy eigenstates for a particle of mass m in a box of length L have wave functions

$\phi_n (x)= \sqrt{\frac{2}{L}} \sin \left(\frac{n\pi x}{L} \right )$

and energies

$E_n=\frac{n^2 \pi^2 x^2}{2mL^2}$

where =1,2,3,... . At time t = 0, the particle is in a state described as follows.

$\psi(t=0)=\frac{1}{\sqrt{14}} (\phi_1+2\phi_2+3\phi_3 )$

Which of the following is a possible result of a measurement of energy for the state

A. 2E₁
B. 5E₁
C. 7E₁
D. 9E₁
E. 14E₁

(GR0177 #44)

Solution:

The energy:

$E_n=\frac{n^2 \pi^2 x^2}{2mL^2} \rightarrow E_1=\frac{\pi^2 x^2}{2mL^2} \rightarrow E_n=n^2 E_1$

So, the possible result of a measurement of energy is a squared quantity:

$\\E_2=2^2 E_1=4E_1 \\E_3=3^2 E_1=9E_1$
... and so on.

The only choice is (D)

Answer: D

Quantum Mechanics - Harmonic Oscillator

Let $|n\rangle$ represent the normalized n^th energy eigenstate of the one-dimensional harmonic oscillator,

$H|n\rangle=\hbar\omega\left(n+\frac{1}{2}\right)|n\rangle$

If $|\psi\rangle$ is a normalized ensemble state that can be expanded as a linear combination

$|\psi\rangle=\frac{1}{\sqrt{14}}|1\rangle-\frac{2}{\sqrt{14}}|2\rangle+\frac{3}{\sqrt{14}}|3\rangle$

of the eigenstates, what is the expectation value of the energy operator in this ensemble state?

A. $\frac{101}{14} \hbar \omega$

B. $\frac{43}{14} \hbar \omega$

C. $\frac{23}{14} \hbar \omega$

D. $\frac{17}{\sqrt{14}} \hbar \omega$

E. $\frac{7}{\sqrt{14}} \hbar \omega$

(GR0177 #45)

Solution:

The energy eigenstates:

$\begin{aligned} H|1\rangle&=\hbar\omega\left(1+\frac{1}{2}\right)|1\rangle = \frac{3}{2}\hbar\omega|1\rangle \\H|2\rangle&=\frac{5}{2}\hbar\omega|2\rangle \\H|3\rangle&=\frac{7}{2}\hbar\omega|3\rangle \end{aligned}$

The expectation value:

$\begin{aligned} \langle H\rangle &=\langle\psi|H|\psi\rangle \\&=\frac{3}{2}\hbar\omega\left( \frac{1}{\sqrt{14}}\right )^2\langle1|1\rangle+\frac{5}{2}\hbar\omega\left(- \frac{2}{\sqrt{14}}\right )^2\langle2|2\rangle+\frac{7}{2}\hbar\omega\left( \frac{3}{\sqrt{14}}\right )^2\langle3|3\rangle \\&=\left( \frac{3}{28} +\frac{20}{28}+\frac{63}{28}\right)\hbar\omega \\&=\frac{86}{28}\hbar\omega =\frac{43}{14}\hbar\omega \end{aligned}$

Answer: B

Quantum Mechanics – Ladder Operator

The operator, $\hat a = \sqrt{\frac{m \omega_0}{2 \hbar}} \big(\hat x+i\frac{\hat p}{m \omega_0}\big)$ when operating on a harmonic energy eigenstate ψ_n with energy E_n, produces another energy eigenstate whose energy is E_n− ħω₀. Which of the following is true?

I. $\hat a$ commutes with the Hamiltonian.

II. $\hat a$ is a Hermitian operator and therefore an observable.

III. The adjoint operator $\hat a^{\dagger}\neq \hat a$

A. I only
B. II only
C. III only
D. I and II only
E. I and III only

(GR9677 #100)

Solution:

I. FALSE
Commutes if [H, a] = 0
But a is a ladder operator, a raises the energy level so that [H, a] = − ħωa

II. FALSE
Hermitian operator if $\hat a=\hat a^{\dagger}$

$\hat a = \sqrt{\frac{m \omega_0}{2 \hbar}} \big(\hat x+i\frac{\hat p}{m \omega_0}\big)$

$\hat a^{\dagger} = \sqrt{\frac{m \omega_0}{2 \hbar}} \big(\hat x-i\frac{\hat p}{m \omega_0}\big)\neq\hat a$

III. TRUE
See II.

Answer: C