Wave Functions of a Trapped Electron — Question 9

Question 9

A simple polynomial trial state nearly reproduces the ground energy

A real trial wave function is $\psi=Ax(L-x)$ inside an infinite well and zero outside. Take A positive. Use $\int_0^Lx^2(L-x)^2dx=L^5/30$ and $\langle K\rangle=(\hbar^2/2m)\int_0^L|\psi\prime|^2dx$.

1. Normalize the state and calculate its mean kinetic energy.

2. Compare with $E_1=\pi^2\hbar^2/(2mL^2)$ and determine the percentage excess.

Question 9 - Solution

A simple polynomial trial state nearly reproduces the ground energy

(a) Method and calculation

$$\boxed{A=\sqrt{30}/L^{5/2}},\quad\psi\prime=A(L-2x).$$$$\int_0^L|\psi\prime|^2dx=A^2L^3/3=10/L^2\Rightarrow\boxed{\langle K\rangle=5\hbar^2/(mL^2)}.$$

(b) Interpretation and further calculation

$$\boxed{\langle K\rangle/E_1=10/\pi^2=1.01321}.$$$$\boxed{100(\langle K\rangle/E_1-1)=1.32118\%}.$$

Physical check

The trial state satisfies the wall conditions but is not an exact eigenfunction. Its mean energy lies above the true ground energy, consistent with the variational bound.