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## Homework Statement

A particle is in a bound state of the infinite square well. It is in a state represented by the following wavefunction, written here at t=0:

ψ(x)= -√(2/3)√(2/L) * sin (3πx/L) + i*√(1/3)√(2/L) * sin (2πx/L)

(a)Write the full time-dependent wavefunction for this state, and calculate the time-dependent probability density.

(b)Use a computer program to plot the probability density at a series of time steps to show how the probability density evolves with time.

## Homework Equations

**Note: I used "h" to represent h-bar below.

Schrodinger equation: -h

^{2}/(2m) * ∂ψ(x,t)/∂x + V(x,t) * ψ(x,t) = ih * ∂ψ(x,t)/∂t

From deriving the time-independent wave function of particle in infinite square well: ψ(x,t)= ψ(x)*ø(t),

with ø(t)=e

^{-i*E*t/h}

Energy of particle in nth state E

_{n}= n

^{2}*π

^{2}*h

^{2}/(2*m*L

^{2})

## The Attempt at a Solution

I know that the ψ(x) given is a linear combination of energy eigenfunctions, the first being from the n=3 state and the second being from the n=2 state of the particle in an infinite square well.

My first instinct was to multiply each individual eigenfunction by the ø(t) that corresponds to said state. I ended up with a wavefunction of:

ψ(x,t)= -√(2/3)√(2/L) * sin (3πx/L) * e

^{-i*E3*t/h}+ i*√(1/3)√(2/L) * sin (2πx/L) * e

^{-i*E2*t/h},

with E3= energy of particle in n=3 state, and E2= energy of particle in n=2 state.

This all seemed fine and dandy. It solved the time-dependent Schrodinger equation, so I was pleased.

Where things got weird was when I calculated the probability density, ψ*ψ. I ended up with the equation:

ψ*ψ= (2/L) * (2/3 * sin

^{2}(3πx/L) + 1/3 * sin

^{2}(2πx/L) + i*√(2)/3 * sin (2πx/L) * sin (3πx/L) * (e

^{i*(E2-E3)*t/h}- e

^{i*(E3-E2)*t/h}))

This is consistent with the problem statement in that ψ(x, 0) = ψ(x); however I feel like the imaginary term in there is wrong. How can the probablility have an imaginary component? I have tried using Euler's formula to transform the e

^{ikt}term but still end up with an imaginary term.

This also leads to the problem of plotting this probability density function. Is it possible to make a plot with an imaginary term? I am using Mathematica but haven't found much information on the subject.

Any insight to where I went wrong or how to continue and deal with the imaginary term would be most appreciated.

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