Consider the following potential energy: Region 1: U(x) = U0 x < 0 Region 2: U(x) = 0 0 < x < L Region 3: U(x) = U0 x > L where U0 > 0. We will consider a particle with energy E such that 0 < E < U0. There are two possible forms for the wave function that might be used to represent the particle: ?(x) = Ai sin kix + Bi cos kix ?(x) = Ai e^kix + Bi e^-kix where i = 1, 2, 3 indicates the three potential regions. a. Sketch this potential energy well. b. Write down wave functions that describe the behavior of the particle in Region 1, Region 2, and Region 3. Use appropriate subscripts to label all parameters. If any of the coefficients Ai or Bi are zero, identify those coefficients and explain why they are equal to zero. c. Sketch the probability distributions you would expect for the ground state and the first excited state. In Region 2, how is the probability distribution of the ground state different from the probability distribution of the first excited state? In Regions 1 and 3, how is the probability distribution of the ground state different from the probability distribution of the first excited state? d. Use the continuity conditions at x = 0 to show how the coefficients of the wave function in Region 2 are related to the coefficients of the wave function in Region 1.
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The potential energy \( U(x) \) is given as: - Region 1: \( U(x) = U_0 \) for \( x < 0 \) - Region 2: \( U(x) = 0 \) for \( 0 < x < L \) - Region 3: \( U(x) = U_0 \) for \( x > L \) The sketch of the potential energy well is as follows: ``` U(x) | | Show more…
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Consider a particle in a three-dimensional harmonic oscillator with potential energy V(x, y, z) = k/2(x^2 + y^2 + z^2). For this particle in this harmonic oscillator potential you are given the value hbar*omega_0 = 6 eV. The wavefunction of this particle at t = 0 is given by the following superposition of the eigenfunctions psi_nx,ny,nz(x,y,z): psi(x,y,z) = 5/6*psi_0,0,0 - 1/2*psi_0,1,1 - sqrt(2)/6*psi_0,0,2 where the psi_nx,ny,nz are the normalized eigenfunctions of this 3-dim harmonic oscillator. a. Show that with the coefficients given in this expansion, the wave function psi(x,y,z) is normalized. b. If a measurement of the energy is made, what are the values of the energy that could be found? Express these in eV. c. i. What is the probability of finding the system in the ground state? ii. What is the probability of finding the system in the 1st excited state? iii. What is the probability of finding the system in the 2nd excited state? iv. What is the probability of finding the system in the 3rd excited state? d. Find the expectation value of the Hamiltonian for this particle in eV.
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One wants to show that the physical state of a (spinless) particle is completely defined by specifying the probability density $\rho(\mathbf{r})=|\psi(\mathbf{r})|^{2}$ and the probability current $\mathbf{J}(\mathbf{r})$ $a,$ Assume the function $\psi(\mathbf{r})$ known and let $\xi(\mathbf{r})$ be its argument: \[ \psi(\mathbf{r})=\sqrt{\rho(\mathbf{r})} \mathrm{e}^{i t \mathbf{r} \mathbf{r}} \] Show that: \[ \mathbf{J}(\mathbf{r})=\frac{\hbar}{m} \rho(\mathbf{r}) \mathbf{V} \xi(\mathbf{r}) \] Deduce that two wave functions leading to the same density $\rho(\mathbf{r})$ and current $\mathbf{J}(\mathbf{r})$ can differ only by a global phase factor. $b$. Given arbitrary functions $\rho(\mathbf{r})$ and $\mathbf{J}(\mathbf{r})$, show that a quantum state $\psi(\mathbf{r})$ can be associated with them only if $\nabla \times v(r)=0,$ where $v(r)=J(r) / \rho(r)$ is the velocity assuciated with the probability fluid. c. Now assume that the particle is submitted to a magnetic fieid $\mathbf{B}(\mathbf{r})=\mathbf{\nabla} \times \mathbf{A}(\mathbf{r})$ $[\text { see chap. } \mathrm{III}, \text { definition }(\mathrm{D}-20) \text { of the probability current in this case }] .$ Show that: \[ \mathbf{J}=\frac{\rho(\mathbf{r})}{m}[\hbar \nabla \xi(\mathbf{r})-q \mathbf{A}(\mathbf{r})] \] and: \[ \nabla \times \mathbf{v}(\mathbf{r})=-\frac{q}{m} \mathbf{B}(\mathbf{r}) \]
The wave functions for the three states with the dot plots shown in Fig. $39-23$, which have $n=2, \ell=1$, and $m_{\ell}=0,+1$, and $-1$, are $$ \begin{array}{r} \psi_{210}(r, \theta)=(1 / 4 \sqrt{2 \pi})\left(a^{-3 / 2}\right)(r / a) e^{-r / 2 a} \cos \theta \\ \psi_{21+1}(r, \theta)=(1 / 8 \sqrt{\pi})\left(a^{-3 / 2}\right)(r / a) e^{-r / 2 a}(\sin \theta) e^{+i \phi}, \\ \psi_{21-1}(r, \theta)=(1 / 8 \sqrt{\pi})\left(a^{-3 / 2}\right)(r / a) e^{-n^{2 a}}(\sin \theta) e^{-i \phi}, \end{array} $$ in which the subscripts on $\psi(r, \theta)$ give the values of the quantum numbers $n, \ell, m_{\ell}$ and the angles $\theta$ and $\phi$ are defined in Fig. $39-22 .$ Note that the first wave function is real but the others, which involve the imaginary number $i$, are complex. Find the radial probability density $P(r)$ for (a) $\psi_{210}$ and (b) $\psi_{21+1}$ (same as for $\psi_{21-1}$ ). (c) Show that each $P(r)$ is consistent with the corresponding dot plot in Fig. $39-23 .$ (d) Add the radial probability densities for $\psi_{210}$, $\psi_{21+1}$, and $\psi_{21-1}$ and then show that the sum is spherically symmetric, depending only on $r$.
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