Q4-1

- -

~ .f' ~ ~ t ~ ( ~ h ~ ) ~ tt-½ i=~ r,'~ l 1~~ :pc.,µ,.,,v, h-~) 4,

Q4b, po is assumed to be 10 times hbar

PlotIfAbs[x] > 1, 0, Cos[10 * x] * 1 - Abs[x],

{x, -3, 3}, AxesLabel → {"x/b", "Re [ψ/A]"}

-3 -2 -1 1 2 3x/b

-0.5

0.5

1.0

Re [ψ/A]

PlotIfAbs[x] > 1, 0, Sin[10 * x] * 1 - Abs[x],

{x, -3, 3}, AxesLabel → {"x/b", "Re [ψ/A]"}

-3 -2 -1 1 2 3x/b

-0.5

0.5

Re [ψ/A]

Q4(c), For plotting, I assume b=10 times hbar

PlotSin10 * p - 1^4 10 * p - 1^4, {p, -3, 3},

AxesLabel → {"p/p0", "Prob / (12/π)"}, PlotRange → All

-3 -2 -1 1 2 3p/p0

0.2

0.4

0.6

0.8

1.0

Prob / (12/π)

Small wings exist in this probability distribution which can be seen if we zoom in a little bit. See graph

below.

PlotSin10 * p - 1^4 10 * p - 1^4,

{p, -3, 3}, AxesLabel → {"p/p0", "Prob / (12/π)"}

-3 -2 -1 1 2 3p/p0

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

Prob / (12/π)

Q7, For plotting, I assume h=1

In[6]:= Pdensity[n_] := 2 * Pi * n^2 p^2 - n^2 * Pi^2^2 * 1 + -1^n + 1 * Cos[p]

2 HW8 QM 2018.nb

In[11]:= Plot[Pdensity[1], {p, -20, 20}, AxesLabel → {"p", "Prob density rescaled, n=1"}]

Out[11]=

-20 -10 10 20p

0.02

0.04

0.06

0.08

0.10

0.12

Prob density rescaled, n=1

In[10]:= Plot[Pdensity[2], {p, -20, 20}, AxesLabel → {"p", "Prob density rescaled, n=2"}]

Out[10]=

-20 -10 10 20p

0.02

0.04

0.06

0.08

Prob density rescaled, n=2

In[12]:= Plot[Pdensity[10], {p, -20, 20}, AxesLabel → {"p", "Prob density rescaled, n=3"}]

Out[12]=

-20 -10 10 20p

0.0005

0.0010

0.0015

0.0020

Prob density rescaled, n=3

HW8 QM 2018.nb 3

Q9

In[14]:= PlotExp-x^2 3, {x, -5, 5}, AxesLabel → {"ψ/A", "x"}

Out[14]=

-4 -2 2 4ψ/A

0.2

0.4

0.6

0.8

1.0

x

In[15]:= PlotExp1 x^2 + 2, {x, -5, 5}, AxesLabel → {"ψ/B", "x"}

Out[15]=

-4 -2 2 4ψ/B

1.1

1.2

1.3

1.4

1.5

1.6

x

In[17]:= Plot[Sech[x / 5], {x, -10, 10}, AxesLabel → {"ψ/C", "x"}]

Out[17]=

-10 -5 5 10ψ/C

0.4

0.6

0.8

1.0

x

We can also use Mathematica to compute probabilities and check whether the function is normalized.

This is done below.

4 HW8 QM 2018.nb

In[20]:= CC = 1 Sqrt[10]

Out[20]=1

10

In[21]:= ψ = CC * Sech[x / 5]

Out[21]=Sech x

5

10

In[22]:= Integrate[ψ^2, {x, -∞, ∞}]

Out[22]= 1

In[23]:= Integrate[ψ^2, {x, 0, 1}]

Out[23]=12Tanh 1

5

In[24]:= N[%]

Out[24]= 0.0986877

Q11In[29]:= Ψ[x_, t_] := 1 Sqrt[I * t + 2] * Exp-x^2 + I * 4 * x - 2 * t 2 * I * t + 2

I animate the real and the imaginary parts and then the absolute value of the wavefunction. See how

the wavefunction spreads as if it is breaking apart. This phenomenon is called dispersion. The intial

wavefunction is NOT an eigenfunction of the Hamiltonian.

HW8 QM 2018.nb 5

In[44]:= Animate[Plot[Re[Ψ[x, t]], {x, -50, 50}, PlotRange → All],

{t, 0, 30}, AnimationRate → .8, AnimationRunning → True]

Out[44]=

t

-40 -20 20 40

-0.4

-0.2

0.2

0.4

6 HW8 QM 2018.nb

In[45]:= Animate[Plot[Im[Ψ[x, t]], {x, -50, 50}, PlotRange → All],

{t, 0, 30}, AnimationRate → .8, AnimationRunning → True]

Out[45]=

t

-40 -20 20 40

-0.2

0.2

0.4

0.6

HW8 QM 2018.nb 7

In[48]:= Animate[Plot[Abs[Ψ[x, t]], {x, -50, 50}, PlotRange → All],

{t, 0, 30}, AnimationRate → .8, AnimationRunning → True]

Out[48]=

t

-40 -20 20 40

0.1

0.2

0.3

0.4

0.5

0.6

8 HW8 QM 2018.nb