# 427: Linear Variational Method for a Particle in a Slanted 1D Box

Trial wavefunctions:

• $$\psi_{1} (x) = \sqrt{2} \sin( \pi x)$$
• $$\psi_{2} (x) = \sqrt{105} x (1-x)^{2}$$

Plot trial wavefunctions and potential energy. x = 0, .005 ... 1 Evaluate matrix elements:

\begin{align} S_{11} &= \int_{0}^{1} \psi _{1} (x)^{2} dx \\[4pt] &= 1 \end{align}

\begin{align} S_{12} &= \int_{0}^{1} \psi _{1} (x) \psi _{2} (x) dx\\[4pt] &= 0.9347 \end{align}

\begin{align} S_{22} &= \int_{0}^{1} \psi _{2} (x)^{2} dx \\[4pt] &= 1 \end{align}

$$H_{11} = \int_{0}^{1} \psi _{1} (x) (- \frac{1}{2}) \frac{d^{2}}{dx^{2}} \psi _{1} (x) dx + \int_{0}^{1} \psi _{1} (x)~x~ \psi _{1} (x) dx$$ $$H_{11} = 5.4348$$

$$H_{12} = \int_{0}^{1} \psi _{1} (x) (- \frac{1}{2}) \frac{d^{2}}{dx^{2}} \psi _{2} (x) dx + \int_{0}^{1} \psi _{1} (x)~x~ \psi _{2} (x) dx$$ $$H_{12} = 5.0163$$

$$H_{22} = \int_{0}^{1} \psi _{2} (x) (- \frac{1}{2}) \frac{d^{2}}{dx^{2}} \psi _{2} (x) dx + \int_{0}^{1} \psi _{2} (x)~x~ \psi _{2} (x) dx$$ $$H_{22} = 7.375$$

Solve the secular equations and normalization constraint for the energy and coefficients.

Seed values for energy and coefficients: E = 5 c1 = .5 c2 = .5

Given

$$(H_{11} - E S_{11})c_{1} + (H_{12} - E S_{12})c_{2} = 0$$

$$(H_{12} - E S_{12})c_{1} + (H_{22} - E S_{22})c_{2} = 0$$

$$c_{1}^{2} S_{11} + 2 c_{1} c_{2} S_{12} + c_{2}^{2} S_{22} = 1$$

$${\begin{pmatrix} E \\ c_{1} \\ c_{2} \end{pmatrix}} = Find(E, c_{1}, c_{2})$$

$${\begin{pmatrix} E \\ c_{1} \\ c_{2} \end{pmatrix}} = {\begin{pmatrix} 5.4328 \\ 0.971\\ 0.031 \end{pmatrix}}$$

Compare variational ground state to PIB ground state: Calculate average position of the particle in the box:

$\int_{0}^{1} x \Phi (x)^{2} dx = 0.496$

Calculate the probability that the particle is in the left half of the box:

$\int_{0}^{0.5} \Phi (x)^{2} dx = 0.5088$