# Homework 3

- Page ID
- 143063

Name: ______________________________

Section: _____________________________

Student ID#:__________________________

## Q1

Which of the following operators are linear?

- \( \hat{A} f(x) = \sqrt{f(x)}\) [square root of f(x)]
- \( \hat{A} x \cdot f(x)= 10 \cdot f(10)\) [evaluate at x=10]
- \( \displaystyle \hat{A} = x^2 \frac{d^2}{dx^2} \)
- \( \hat{A} f(x) =\log_{10} f(x) \) [take the log of f(x)]
- \( \hat{A} f(x) =\cos f(x) \) [take the cos of f(x)]
- \( \hat{A} f(x) = 10 \) [multiply f(x) by ten]
- \( y \cdot \hat{ A} f(x) = f(x )\) [where y is not a constant]
- \( \hat{A} f(x) = 0 \) [multiply f(x) by zero]
- \( \hat{A} f(x) =f^∗(x)\) [form the complex conjugate of f(x)]
- \( \hat{A} = \left( \begin{array}{cc} a & b \\ d & e \end{array} \right) \)
- \( \hat{H} = \left( \begin{array}{cc} 1 & \delta \\ \delta & 1 \end{array} \right) \)
- \( \displaystyle \hat{A} f(x) = \int_0^x f(x')dx' \) [Integrating from 0 to x of f(x)]
- \( \hat{A} f(x) =[[f(x)]^{−1}]^{−1} \) [take the reciprocoal of the reciprocal of f(x)]
- \( \displaystyle \hat{A} f(x,y) = \dfrac{\partial f(x,y)}{\partial x} + \dfrac{\partial f(x,y)}{\partial y} \) [sum of two partial derivatives f(x,y)]

## Q2

Demonstrate that the function \(e^{-ikx}\) is an eigenfunction of the kinetic energy operator. What are the corresponding eigenvalues?

## Q3

Given a particle in a 1D box with infinitely high walls in the \(n=3\) state:

- Draw or plot the wavefunction.
- Draw or plot the probability. (Wavefunction^2)
- How many nodes are there (the boundaries do not count)?
- How many antinodes are there?
- What is the probability of finding the particle outside the box?
- How do the above questions (c,d,e) change if:
- The mass is doubled (i.e., \(2m\))?
- The box length is doubled (i.e., \(2L\))?
- The quantum number is doubled (i.e., \(2n\))?

## Q4

Find following values for a particle of mass, \(m \), for a particle in a 1D box with length \(L\) in the \(n=1\) state and \(n=2\) states.

- \(\langle x \rangle\)
- \(\langle x^2 \rangle\)
- \(\langle p \rangle\)
- \(\langle p^2 \rangle\)

## Q5

The uncertainties of a position and a momentum of a particle (\(\Delta x\)) and (\(\Delta p\)) are defined as

\[ \Delta x = \sqrt{ \langle x^2 \rangle - \langle x \rangle ^2} \]

\[ \Delta p = \sqrt{ \langle p^2 \rangle - \langle p \rangle ^2} \]

- For the particle in the box at the ground eigenstate (\(n=1\)) and first excited state (\(n=2\)), what is the uncertainty in the value \( x \)? How would you interpret the results of these calculations? (Hint: Do Q4 first and use solutions from Q4.)
- For the particle in the box at the ground eigenstate (\(n=1\)) and first excited state (\(n=2\)), what is the uncertainty in the value \( p \)? How would you interpret the results of these calculations?
- What is the product for the ground and first excited state: \(\Delta x \Delta p\).
- Does the Heisenberg Uncertainty Principle hold for a particle in each of these states?

## Q6

What is the *expectation value* of kinetic energy for a particle in a box of length (\(L\)) in the ground eigenstate (n=1)? What about for the third excited eignestate (n=3). Explain the difference.

## Q7

Consider a particle of mass \(m\) in a one-dimensional box of width \(L\). Calculate the general formula for energy of the transitions between neighboring states. How much energy is required to excite the particle from the \(n=1\) to \(n=2\) state?

## Q8

Consider a particle of mass \(m\) in a two-dimensional square box with sides \(L\). Calculate the three lowest (__different!__) energies of the system. Write them down in the increasing order with their principal quantum numbers.

## Q9

For a particle in a one-dimensional box of length \(L\), the second excited state wavefunction (n=5) is

\[\psi_5=\sqrt{\dfrac{2}{L}}\sin{\dfrac{5\pi x}{L}}\]

- What is the probability that the particle is in the left half of the box?
- What is the probability that the particle is in the middle third of the box?

## Q10

For each wavefunction below, (i) normalize the following wavefunction (if possible). (ii) sketch a plot of \( |\psi |^2 \) as a function of \( x \).

- \( \psi(x) = Ae^{-|x|/a_o} \) with \(A,a_o \) all real
- \( \psi(x) = A \sin (\pi x/2a) \) restricted to the region \(-a < x < a \)
- \( \psi(x) = A \cos (\pi x/2a) \) restricted to the region \(-a < x < a \)
- \( \psi(x,t) = Ae^{i(\omega t - kx)} \) with \(A,k,\omega\) all real