Question 1

Which equation represents the time-independent Schrödinger equation for a particle in a three-dimensional box?

Accepted Answer

$$-\frac{\$$hbar^2$$}{2m} \left( \frac{\$$partial^2$$ \psi}{\partial x^2$} + \frac{\$$partial^2$$ \psi}{\partial y^2$} + \frac{\$$partial^2$$ \psi}{\partial z^2$} \right) + U(x,y,z) \psi = E \psi$$

Question 2

For a particle in a three-dimensional cubical box, which of the following is the correct expression for the energy levels?

Accepted Answer

$$E = \frac{\pi^2$ \$$hbar^2$$}{2mL^2$} (n_X^2$ + n_Y^2$ + n_Z^2$)$$

Question 3

What is the normalization condition for a stationary state in three dimensions?

Accepted Answer

$$\int_{-\infty}^{+\infty} |\psi(x,y,z)|^2 dV = 1$$

Question 4

Given the wave function $$\psi_{2,1,1}(x,y,z)$$ for a particle in a cubical box, what is the probability distribution function zero?

Accepted Answer

On the plane $$x = \frac{L}{2}$$

Question 5

Which quantum numbers correspond to the wave function $$\psi_{1,2,1}(x,y,z)$$?

Accepted Answer

$$n_X = 1, n_Y = 2, n_Z = 1$$

Question 6

What is the degeneracy of the energy level $$E = 2E_{1,1,1}$$ for a particle in a cubical box?

Accepted Answer

3

Question 7

If the box is made asymmetric such that $$L_X 
eq L_Y 
eq L_Z$$, what happens to the degeneracy of the energy levels?

Accepted Answer

Degeneracy is removed

Question 8

Which of the following is a valid form for the wave function of a particle in a three-dimensional box?

Accepted Answer

$$\psi(x,y,z) = C \sin\left(\frac{n_X \pi x}{L}\right) \sin\left(\frac{n_Y \pi y}{L}\right) \sin\left(\frac{n_Z \pi z}{L}\right)$$

Question 9

For the energy level $$E = 3E_{1,1,1}$$, which set of quantum numbers is NOT associated with this energy?

Accepted Answer

$$(3,1,1)$$

Question 10

A particle in a three-dimensional box has quantum numbers $$n_X = 1$$, $$n_Y = 1$$, $$n_Z = 2. $$Where is the probability distribution function zero?

Accepted Answer

On the plane $$z = \frac{L}{2}$$

Question 11

Which of the following statements about the energy levels in a three-dimensional box is correct?

Accepted Answer

Energy levels depend on the sum of the squares of the quantum numbers

Question 12

If a particle is in the state $$\psi_{1,1,1}$$, what is the energy of the particle?

Accepted Answer

$$E_{1,1,1} = \frac{\pi^2$ \$$hbar^2$$}{2mL^2$}$$

Question 13

Which of the following is a consequence of symmetry in a cubical box?

Accepted Answer

Degeneracy of energy levels

Question 14

For a particle in a three-dimensional box, which quantum numbers correspond to the lowest possible energy state?

Accepted Answer

$$n_X = 1, n_Y = 1, n_Z = 1$$

Question 15

Which of the following best describes the separation of variables technique used for solving the Schrödinger equation in three dimensions?

Accepted Answer

Expressing the wave function as a product of functions, each depending on a single coordinate

Question 16

If the quantum numbers for a particle in a box are $$n_X = 3$$, $$n_Y = 1$$, $$n_Z = 1$$, what is the energy of the particle?

Accepted Answer

$$E = \frac{11}{3} E_{1,1,1}$$

Question 17

Which of the following is NOT a possible quantum number for a particle in a three-dimensional box?

Accepted Answer

$$n_X = 0$$

Question 18

If a particle in a three-dimensional box is in the state $$\psi_{2,2,2}$$, what is the energy of the particle?

Accepted Answer

$$E = 4 E_{1,1,1}$$

Quantum Mechanics in Three Dimensions: Particle in a Box

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