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)]
- \( \hat{A} f(x) =\sin f(x) \) [take the sin of f(x)]
- \( \hat{A} f(x) = 10 \) [multiply f(x) by ten]
- \( \hat{A} f(x) = 0 \) [multiply f(x) by zero]
- \( \hat{A} f(x) =f^∗(x)\) [form the complex conjugate of f(x)]
- \( \hat{A} f(x) =f^∗(ix)\) [form the complex conjugate of f(ix)]
- \( \hat{A} = \left( \begin{array}{cc} a & b \\ d & e \end{array} \right) \)
- \( \hat{H} = \left( \begin{array}{cc} 1 & \delta \\ \delta & 1 \end{array} \right) \)
- \( \displaystyle \hat{A} f(x) = \int_0^x f(x')dx' \) [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
For the operations below, given \( f(x) = x \), (a) Is this an eigenfunction? (b) determine the eigenvalue if it is an eigenfunction.
- \( \hat{A} f(x) = \sqrt{f(x)}\) [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]
- \( \hat{A} f(x) = 0 \) [multiply f(x) by zero]
- \( \hat{A} f(x) =f^∗(x)\) [form the complex conjugate of f(x)]
- \( \displaystyle \hat{A} f(x) = \int_0^x f(x')dx' \) [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)]
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:
- 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., \(2n\))?
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.
- \(\langle x \rangle\)
- \(\langle x^2 \rangle\)
- \(\langle p \rangle\)
- \(\langle p^2 \rangle\)
Q6
The uncertainties of a position and a momentum of a particle (\(\Delta x\)) and (\(\Delta p\)) are defined as
\[ \Delta x = \sqrt{ \langle x^2 \rangle - \langle x \rangle ^2} \]
\[ \Delta p = \sqrt{ \langle p^2 \rangle - \langle p \rangle ^2} \]
- 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?
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
\[\psi_6=\sqrt{\dfrac{2}{L}}\sin{\dfrac{6\pi x}{L}}\]
- 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?
Q11
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
- \( \psi(x) = A \sin (\pi \frac{x}{2a}) \) restricted to the region \(-a < x < a \)
- \( \psi(x) = A \cos (\pi \frac{x}{2a} \) restricted to the region \(-a < x < a \)
- \( \psi(x,t) = Ae^{i(\omega t - kx)} \) with \(A,k,\omega\) all real