Homework 3

Name: ______________________________

Section: _____________________________

Student ID#:__________________________

Q1

Write the time-independent and time-dependent Schrödinger equations for each the following systems:

1. A particle of mass \(m\) moving in 3-D space with the potential energy given by \(V(x, y, z) = -\dfrac {(x^2 + y^2 + z^2)}{3}\).
2. A particle of mass \(mu\) moving in 1-D space with a potential energy given by \(V(x)=\frac{1}{2} k (x-x_o)^2\) where \(x_o\) and \(k\) are constants.
3. A particle of mass \(m\) within a 1-D box with infinitely high walls and of length \(L\).
4. A particle of mass \(m\) within a 1-D box with finite high walls of length \(L\) and height \(B\).

Q2

Which of the following operators are linear?

1. \( \displaystyle \hat{A} = 2x^2 \frac{d^2}{dx^2} \)
2. \( \displaystyle \hat{A} f(x) = f(x) \)
3. \( \displaystyle \hat{A} f(x,y) = \dfrac{\partial f(x,y)}{\partial x} + \dfrac{\partial f(x,y)}{\partial y} \) [sum of two partial derivatives f(x,y)]
4. \( \hat{A} f(x) = \sqrt{f(x)}\) [square f(x)]
5. \( \hat{A} f(x) =f^∗(x)\) [form the complex conjugate of f(x)]
6. \( \hat{A} = \left( \begin{array}{cc} a & b \\ d & e \end{array} \right) \)
7. \( \hat{H} = \left( \begin{array}{cc} 1 & \delta \\ \delta & 1 \end{array} \right) \)
8. \( \displaystyle \hat{A} f(x) = \int_0^x f(x')dx' \) [Integrating from 0 to x of f(x)]
9. \( \hat{A} f(x) = 0 \) [multiply f(x) by zero]
10. \( \hat{A} f(x) =[f(x)]^{−1} \) [take the reciprocal of f(x)]
11. \( \hat{A} f(x)=f(0)\) [evaluate f(x) at x=0]
12. \( \hat{A} f(x) =\log_{10} f(x) \) [take the log of f(x)]
13. \( \hat{A} f(x) =\sin ⁡f(x) \) [take the sin of f(x)]

Q3

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

Q4

Given a particle in a 1D box with infinite high walls in the \(n=5\) state:

1. How many nodes are there (the edges do not count)?
2. How many antinodes are there?
3. What is the probability of finding the particle outside the box?
4. How do the above questions change if:
   1. The mass is doubled (i.e., \(2m\))?
   2. The box length is doubled (i.e., \(2L\))?
   3. The quantum number is doubled (i.e., \(2n\))?

Q5

Consider a particle of mass \(m\) that has energy \( E = 0 \) and a time-independent wave function given by:

\[ \psi(x) = A x^2 e^{-B^2 x^2} \]

\(A \) and \(B\) are constants. Determine the potential energy \( V(x) \) of the particle. (Hint: Use the time-independent Schroedinger equation.) What does this potential energy look like?

Q6

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 \(n=3\) states.

1. \(\langle x \rangle\)
2. \(\langle x^2 \rangle\)
3. \(\langle p \rangle\)
4. \(\langle p^2 \rangle \)

Q7

For each of the expectation values in Q6, explain why or why not, the respective values depends on \(n\).

Q8

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

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

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

1. 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?
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 \( p \)? How would you interpret the results of these calculations?
3. What is the product for the ground and first excited state: \(\Delta x \Delta p\).
4. Does the Heisenberg Uncertainty Principle hold for a particle in each of these states?

Q9

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 first excited eignestate (n=2). Explain the difference.

Q10

What is the most probable position for a particle in a box of length (\(L\)) in the ground eigenstate (n=1)? What about for the first excited eignestate (n=2). Explain the difference.

Q11

Find the following expectation values for a particle with mass, \(m \), in a 3D box with dimensions \(a_1 \), \(a_2 \), \(a_3 \). Assume that the quantum numbers are given by \(n_x\), \(n_y\), and \(n_z\).

1. \( \langle x \rangle \)
2. \( \langle y \rangle \)
3. \( \langle z \rangle \)
4. \( \langle p_z \rangle \)
5. \( \langle z^2 \rangle \)
6. \( \langle z \rangle^2 \)

Q12

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 \(n=2\) to \(n=3\) state?

Q13

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

Q14

For a particle in a one-dimensional box of length \(L\), the second excited state wavefunction (n=3) is

\[\psi_3=\sqrt{\dfrac{2}{L}}\sin{\dfrac{3\pi x}{L}}\]

1. What is the probability that the particle is in the left half of the box?
2. What is the probability that the particle is in the middle third of the box?

Q15

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

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