Ex 2.1-1 Particle in a Box

Remove["Global`*"]

hbar := \[HBar]

H[V_] @ psi_  := -hbar^2/(2m) D[psi,{x,2}] + V psi

SchD[V_] @  psi_ := H[V] @ psi - En psi

SchD[V[x]] @ psi[x] == 0 // TeXForm

\displaystyle{-\text{En} \psi (x)-\frac{\hbar ^2 \psi ''(x)}{2 m}+\psi (x) V(x)=0}

phi[n_, x_] := Cn Sin[n Pi x/L]

SchD[0] @ phi[n,x] == 0 // ExpandAll

SchD[0] @ phi[n,x] / phi[n,x] == 0 // Simplify // TeXForm

\displaystyle{2 \text{En} L=\frac{\pi ^2 n^2 \hbar ^2}{L m}}

Starting with a linear superposition A E^(Ikx) + B E^(-Ikx) of independent plane waves, where A and B are constants, verify the box eigen-functions and eigen-energies given above.

Thus, show that this superposition is a solution of the Schrodinger equation and, by invoking the boundary condition, that k -> n Pi/L.

— Quantum Methods with Mathematica

phi[k_, x_] := A E^(I k x) + B E^(- I k x)

eq1 := SchD[0] @ phi[k,x]/phi[k,x] == 0 // Simplify

Ek[k_] := En /. Solve[eq1, En] [[1]]

Ek[k] // TeXForm

\displaystyle{\frac{k^2 \hbar ^2}{2 m}}

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The boundary conditions are \displaystyle{\phi_{x=0} = \phi_{x=L} = 0}.

\displaystyle{  \begin{aligned}  A+B&=0 \\   A e^{i k L}+B e^{-i k L}&=0 \\  \end{aligned}}

So

\displaystyle{  \begin{aligned}  e^{i k L} - e^{-i k L}&=0 \\  \end{aligned}}

Solve[E^(I k L) - E^(-I k L) == 0, k] // Simplify

\displaystyle{  \left\{k\to -\frac{2 \pi  c_1}{L}\text{ if }c_1\in \mathbb{Z}, ~~~k\to -\frac{2 \pi  c_1+\pi }{L}\text{ if }c_1\in \mathbb{Z} \right\}}

\displaystyle{  =\left\{k\to \frac{n \pi }{L}\text{ if }n \in \mathbb{Z}\right\}}

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phi[n_, x_] := Cn Sin[n Pi x/L]

norm[n_] = Cn /.
    Solve[
        Integrate[ phi[n,x]^2, {x,0,L}] == 1 /.
            Sin[m_Integer n Pi] -> 0, 
        Cn
    ] [[1]] // TeXForm

\displaystyle{  -\frac{\sqrt{2}}{\sqrt{L}}  }

It’s interesting to note in passing that the 1D box eigenfunctions are also classically the eigenfunctions of a taut string. However, whereas the quantum mechanical energies scale as n^2, the classical eigenfrequencies of the string’s normal modes are linear in n. This is a consequence of the classical wave equation being second order in time in contrast to the quantum wave equation being first order.

— Quantum Methods with Mathematica

— Me@2022-12-16 10:23:45 AM

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2022.12.17 Saturday (c) All rights reserved by ACHK