Question

Consider the following example: The interface between the semiconductors GaAs and Al$_{1-x}$Ga$_x$As forms what is known as in inversion layer, in which electrons are trapped in a triangular potential well. For the purposes of this problem, assume that the potential is\\ $V = \infty$, for $x \leq 0$;\\ $V = Ex$, for $x \geq 0$.\\In other words, this is a triangular potential well with an infinite wall at $x = 0$.\\(a) Look up the exact solution in a reference of your choice. What is the ground state energy?\\(b) I claim that the function\\ $\psi(x) = 0$, $x \leq 0$\\ $\psi(x) = Cxe^{-ax}$, $x \geq 0$\\ Is a reasonable guess for a wavefunction. Considering the boundary conditions and limiting behavior as $x \to \infty$, explain why this choice is reasonable.\\(c) Given that the wave-function must be normalized, what is C?\\(d) Use the variational principle to determine $a$.\\(e) What is the estimate for the ground state energy that you obtain? Compare with the exact solution.

          Consider the following example: The interface between the semiconductors GaAs and Al$_{1-x}$Ga$_x$As forms what is known as in inversion layer, in which electrons are trapped in a triangular potential well. For the purposes of this problem, assume that the potential is\\
$V = \infty$, for $x \leq 0$;\\
$V = Ex$, for $x \geq 0$.\\In other words, this is a triangular potential well with an infinite wall at $x = 0$.\\(a) Look up the exact solution in a reference of your choice. What is the ground state energy?\\(b) I claim that the function\\
$\psi(x) = 0$, $x \leq 0$\\
$\psi(x) = Cxe^{-ax}$, $x \geq 0$\\
Is a reasonable guess for a wavefunction. Considering the boundary conditions and limiting behavior as $x \to \infty$, explain why this choice is reasonable.\\(c) Given that the wave-function must be normalized, what is C?\\(d) Use the variational principle to determine $a$.\\(e) What is the estimate for the ground state energy that you obtain? Compare with the exact solution.

Consider the following example: The interface between the semiconductors GaAs and Al1-xGaxAs forms what is known as in inversion layer, in which electrons are trapped in a triangular potential well. For the purposes of this problem, assume that the potential is

V = ∞, for x ≤ 0;

V = Ex, for x ≥ 0.
In other words, this is a triangular potential well with an infinite wall at x = 0.
(a) Look up the exact solution in a reference of your choice. What is the ground state energy?
(b) I claim that the function

ψ(x) = 0, x ≤ 0

ψ(x) = Cxe^-ax, x ≥ 0

Is a reasonable guess for a wavefunction. Considering the boundary conditions and limiting behavior as x →∞, explain why this choice is reasonable.
(c) Given that the wave-function must be normalized, what is C?
(d) Use the variational principle to determine a.
(e) What is the estimate for the ground state energy that you obtain? Compare with the exact solution.

Added by Inmaculada P.

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