Question

2.32 A particle of mass $m$ is confined to a one-dimensional square well with finite walls. That is, a potential $V(x) = 0$ for $-a \le x \le +a$, and $V(x) = V_0 = \eta(h^2/2ma^2)$ otherwise. You are to find the bound-state energy eigenvalues as $E = \varepsilon V_0$ along with their wave functions. a. Set the problem up with a wave function $Ae^{\alpha x}$ for $x \le -a$, $De^{-\alpha x}$ for $x \ge +a$, and $Be^{ikx} + Ce^{-ikx}$ inside the well. Match the boundary conditions at $x = \pm a$ and show that $k$ and $\alpha$ must satisfy $z = \pm z^*$ where $z = e^{i\alpha k}(k - i\alpha)$. Proceed to find a purely real or purely imaginary expression for $z$ in terms of $k$ and $\alpha$. b. Find the wave functions for the two choices of $z$ and show that the purely real (imaginary) choice leads to a wave function that is even (odd) under the exchange $x \to -x$. c. Find a transcendental equation for each of the two wave functions relating $\eta$ and $\varepsilon$. Show that even a very shallow well ($\eta \to 0$) has at least one solution for the even wave function, but you are not guaranteed any solution for an odd wave function. d. For $\eta = 10$, find all the energy eigenvalues and plot their normalized wave functions.

          2.32 A particle of mass $m$ is confined to a one-dimensional square well with finite walls. That is, a potential $V(x) = 0$ for $-a \le x \le +a$, and $V(x) = V_0 = \eta(h^2/2ma^2)$ otherwise. You are to find the bound-state energy eigenvalues as $E = \varepsilon V_0$ along with their wave functions.
a. Set the problem up with a wave function $Ae^{\alpha x}$ for $x \le -a$, $De^{-\alpha x}$ for $x \ge +a$, and $Be^{ikx} + Ce^{-ikx}$ inside the well. Match the boundary conditions at $x = \pm a$ and show that $k$ and $\alpha$ must satisfy $z = \pm z^*$ where $z = e^{i\alpha k}(k - i\alpha)$. Proceed to find a purely real or purely imaginary expression for $z$ in terms of $k$ and $\alpha$.
b. Find the wave functions for the two choices of $z$ and show that the purely real (imaginary) choice leads to a wave function that is even (odd) under the exchange $x \to -x$.
c. Find a transcendental equation for each of the two wave functions relating $\eta$ and $\varepsilon$. Show that even a very shallow well ($\eta \to 0$) has at least one solution for the even wave function, but you are not guaranteed any solution for an odd wave function.
d. For $\eta = 10$, find all the energy eigenvalues and plot their normalized wave functions.

2.32 A particle of mass m is confined to a one-dimensional square well with finite walls. That is, a potential V(x) = 0 for -a ≤ x ≤ +a, and V(x) = V0 = η(h^2/2ma^2) otherwise. You are to find the bound-state energy eigenvalues as E = ε V0 along with their wave functions.
a. Set the problem up with a wave function Ae^α x for x ≤ -a, De^-α x for x ≥ +a, and Be^ikx + Ce^-ikx inside the well. Match the boundary conditions at x = ± a and show that k and α must satisfy z = ± z^* where z = e^iα k(k - iα). Proceed to find a purely real or purely imaginary expression for z in terms of k and α.
b. Find the wave functions for the two choices of z and show that the purely real (imaginary) choice leads to a wave function that is even (odd) under the exchange x → -x.
c. Find a transcendental equation for each of the two wave functions relating η and ε. Show that even a very shallow well (η→ 0) has at least one solution for the even wave function, but you are not guaranteed any solution for an odd wave function.
d. For η = 10, find all the energy eigenvalues and plot their normalized wave functions.

Added by Steven G.

Question

Please give Ace some feedback