Homework 3

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Q1

Which of the following operators are linear?

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

Q2

For the operations below, given \( f(x) = x \), (a) Is this an eigenfunction? (b) determine the eigenvalue if it is an eigenfunction.

1. \( \hat{A} f(x) = \sqrt{f(x)}\) [square root of f(x)]
2. \( \hat{A} x \cdot f(x)= 10 \cdot f(10)\) [evaluate at x=10]
3. \( \displaystyle \hat{A} = x^2 \frac{d^2}{dx^2} \)
4. \( \hat{A} f(x) =\log_{10} f(x) \) [take the log of f(x)]
5. \( \hat{A} f(x) =\cos ⁡f(x) \) [take the cos of f(x)]
6. \( \hat{A} f(x) = 10 \) [multiply f(x) by ten]
7. \( \hat{A} f(x) = 0 \) [multiply f(x) by zero]
8. \( \hat{A} f(x) =f^∗(x)\) [form the complex conjugate of f(x)]
9. \( \displaystyle \hat{A} f(x) = \int_0^x f(x')dx' \) [Integrating from 0 to x of f(x)]
10. \( \hat{A} f(x) =[[f(x)]^{−1}]^{−1} \) [take the reciprocoal of the reciprocal of f(x)]
11. \( \displaystyle \hat{A} f(x,y) = \dfrac{\partial f(x,y)}{\partial x} + \dfrac{\partial f(x,y)}{\partial y} \) [sum of two partial derivatives f(x,y)]

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 infinitely high walls in the \(n=3\) state:

1. Draw or plot the wavefunction.
2. Draw or plot the probability. (Wavefunction^2)
3. How many nodes are there (the boundaries do not count)?
4. How many antinodes are there?
5. What is the probability of finding the particle outside the box?
6. How do the above questions (c,d,e) 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

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.

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

Q6

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? (Hint: Do Q4 first and use solutions from Q4.)
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?

 

Q7

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

Q8

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? (Hint: We did a similar problem in class but with numbers.)

Q9

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.

Q10

For a particle in a one-dimensional box of length \(L\), the sixth excited state wavefunction (n=6) is

\[\psi_6=\sqrt{\dfrac{2}{L}}\sin{\dfrac{6\pi x}{L}}\]

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?

Q11

For each wavefunction below, (i) normalize the following wavefunction (if possible). (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. \( \psi(x) = A \sin (\pi \frac{x}{2a}) \) restricted to the region \(-a < x < a \)
3. \( \psi(x) = A \cos (\pi \frac{x}{2a} \) restricted to the region \(-a < x < a \)
4. \( \psi(x,t) = Ae^{i(\omega t - kx)} \) with \(A,k,\omega\) all real