Homework 3 (Larsen)
- Page ID
- 109905
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Name: ______________________________
Section: _____________________________
Student ID#:__________________________
Template:HideTOCQ1
Write the time-independent and time-dependent Schrödinger equations for each the following systems:
- A particle of mass \(m\) moving in 3-D space with the potential energy given by \(V(x, y, z) = -\dfrac {(x^2 + y^2 + z^2)}{3}\).
- 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.
- A particle of mass \(m\) within a 1-D box with infinitely high walls and of length \(L\).
- 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?
- \( \displaystyle \hat{A} = 2x^2 \frac{d^2}{dx^2} \)
- \( \displaystyle \hat{A} f(x) = f(x) \)
- \( \displaystyle \hat{A} f(x,y) = \dfrac{\partial f(x,y)}{\partial x} + \dfrac{\partial f(x,y)}{\partial y} \) [sum of two partial derivatives f(x,y)]
- \( \hat{A} f(x) = \sqrt{f(x)}\) [square f(x)]
- \( \hat{A} f(x) =f^∗(x)\) [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)]
- \( \hat{A} f(x) = 0 \) [multiply f(x) by zero]
- \( \hat{A} f(x) =[f(x)]^{−1} \) [take the reciprocal of f(x)]
- \( \hat{A} f(x)=f(0)\) [evaluate f(x) at x=0]
- \( \hat{A} f(x) =\log_{10} f(x) \) [take the log of f(x)]
- \( \hat{A} f(x) =\sin f(x) \) [take the sin of f(x)]
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:
- How many nodes are there (the edges do not count)?
- How many antinodes are there?
- What is the probability of finding the particle outside the box?
- How do the above questions 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
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.
- \(\langle x \rangle\)
- \(\langle x^2 \rangle\)
- \(\langle p \rangle\)
- \(\langle p^2 \rangle \)
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
\[ \Delta x = \sqrt{ \langle x^2 \rangle - \langle x \rangle ^2} \]
\[ \Delta p = \sqrt{ \langle p^2 \rangle - \langle p \rangle ^2} \]
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?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?
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 eigenstate (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 \(n_z\).
\( \langle x \rangle \)\( \langle y \rangle \)\( \langle z \rangle \)\( \langle p_z \rangle \)\( \langle z^2 \rangle \)\( \langle z \rangle^2 \)
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
\[\psi_3=\sqrt{\dfrac{2}{L}}\sin{\dfrac{3\pi x}{L}}\]
- 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?
Q15
For each wavefunction below, (i) normalize the following wavefunction. (ii) sketch a plot of \( |\psi |^2 \) as a function of x.
- \( \psi(x) = Ae^{-|x|/a_o} \) with \(A,a_o \) all real
- \( \psi(x) = A \cos (\pi x/2a) \) restricted to the region \(-a < x < a \)
- \( \psi(x,t) = Ae^{i(\omega t - kx)} \) with \(A,k,\omega\) all real
- \( \psi(x,t) = Ae^{-\lambda |x|} e^{i(\omega t)} \) with \(A,\lambda,\omega \) all real