Question 1

Which of the following is the correct time-dependent Schrödinger equation for a free particle in one dimension?

Accepted Answer

$$i \hbar \frac{\partial \Psi(x,t)}{\partial t} = -\frac{\$$hbar^2$$}{2m} \frac{\$$partial^2$$ \Psi(x,t)}{\partial x^2$}$$

Question 2

The energy operator (Hamiltonian) in quantum mechanics is represented as:

Accepted Answer

$$\hat{H} = -\frac{\$$hbar^2$$}{2m} \$$nabla^2$$ + V(\vec{r}, t)$$

Question 3

For a particle in a one-dimensional box of length $$L$$, the normalized wave function for the ground state is:

Accepted Answer

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

Question 4

Which of the following is the correct commutation relation between position and momentum operators in one dimension?

Accepted Answer

$$[\hat{x}, \hat{p}_x] = i \hbar$$

Question 5

The expectation value of position $$\langle x \rangle$$ for a normalized wave function $$\psi(x) is $$given by:

Accepted Answer

$$\langle x \rangle = \int \psi^*(x) x \psi(x) dx$$

Question 6

If $$\psi(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{\pi x}{L}\right)$$ for $$0 \u003c x \u003c L$$, what is the expectation value of momentum $$\langle p \rangle$$?

Accepted Answer

$$\langle p \rangle = 0$$

Question 7

Which of the following statements about stationary states is correct?

Accepted Answer

The probability density $$|\psi(x,t)|^2 is $$independent of time.

Question 8

The operator associated with the observable of momentum in the $$x-$$direction is:

Accepted Answer

$$\hat{p}_x = -i \hbar \frac{\partial}{\partial x}$$

Question 9

Which of the following is the correct form of the time-independent Schrödinger equation in one dimension?

Accepted Answer

$$-\frac{\$$hbar^2$$}{2m} \frac{d^2$ \psi(x)}{dx^2$} + V(x) \psi(x) = E \psi(x)$$

Question 10

If $$\hat{A}$$ and $$\hat{B}$$ are operators, the commutator $$[\hat{A}, \hat{B}] is $$defined as:

Accepted Answer

$$[\hat{A}, \hat{B}] = \hat{A}\hat{B} - \hat{B}\hat{A}$$

Question 11

Which of the following is a property of eigenfunctions corresponding to different eigenvalues?

Accepted Answer

They are orthogonal.

Question 12

The expectation value of an observable $$\hat{A}$$ for a normalized wave function $$\psi is $$given by:

Accepted Answer

$$\langle A \rangle = \int \psi^* \hat{A} \psi dx$$

Question 13

For a particle in a 1D box, the energy levels are given by:

Accepted Answer

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

Question 14

Which of the following is the correct normalization condition for a wave function $$\psi(x) in $$one dimension?

Accepted Answer

$$\int_{-\infty}^{\infty} |\psi(x)|^2 dx = 1$$

Question 15

The commutator $$[\hat{x}, \hat{p}_y] is$$:

Accepted Answer

$$0$$

Question 16

If $$\psi(x) = N e$$^{i k x}$$, what is the expectation value $$\langle x \rangle$$?

Accepted Answer

$$0$$

Question 17

Which of the following is the correct operator for kinetic energy in quantum mechanics?

Accepted Answer

$$\hat{T} = -\frac{\$$hbar^2$$}{2m} \$$nabla^2$$

Question 18

The eigenvalue equation $$\hat{A} \psi = \lambda \psi$$ means:

Accepted Answer

$$\psi is an $$eigenfunction of $$\hat{A}$$ with eigenvalue $$\lambda$$

Question 19

Which of the following is true about the probability density for a stationary state?

Accepted Answer

It does not depend on time.

Question 20

The commutator $$[\hat{x}, \hat{p}_x^2$] is$$:

Accepted Answer

$$2i \hbar \hat{p}_x$$

Study Notes: Schrödinger Wave Equation and Quantum Operators

Study Guide - Practice Questions

Study Guide - Flashcards