tt-½ l :pc.,µ,.,,v, · 2018. 12. 10. · Q4(c), For plotting, I assume b=10 times hbar Plot Sin...

42
Q4-1 -- . f' t ( h ) tt-½ i=~ r,'~ l 1 ~~ :pc.,μ,.,,v, h-~) 4,

Transcript of tt-½ l :pc.,µ,.,,v, · 2018. 12. 10. · Q4(c), For plotting, I assume b=10 times hbar Plot Sin...

Page 1: tt-½ l :pc.,µ,.,,v, · 2018. 12. 10. · Q4(c), For plotting, I assume b=10 times hbar Plot Sin 10* p-1 ^4 10* p-1 ^4, {p, -3, 3}, AxesLabel→ {"p/p0", "Prob / (12/π)"}, PlotRange→

Q4-1

- -

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

Page 2: tt-½ l :pc.,µ,.,,v, · 2018. 12. 10. · Q4(c), For plotting, I assume b=10 times hbar Plot Sin 10* p-1 ^4 10* p-1 ^4, {p, -3, 3}, AxesLabel→ {"p/p0", "Prob / (12/π)"}, PlotRange→
Page 3: tt-½ l :pc.,µ,.,,v, · 2018. 12. 10. · Q4(c), For plotting, I assume b=10 times hbar Plot Sin 10* p-1 ^4 10* p-1 ^4, {p, -3, 3}, AxesLabel→ {"p/p0", "Prob / (12/π)"}, PlotRange→
Page 4: tt-½ l :pc.,µ,.,,v, · 2018. 12. 10. · Q4(c), For plotting, I assume b=10 times hbar Plot Sin 10* p-1 ^4 10* p-1 ^4, {p, -3, 3}, AxesLabel→ {"p/p0", "Prob / (12/π)"}, PlotRange→
Page 5: tt-½ l :pc.,µ,.,,v, · 2018. 12. 10. · Q4(c), For plotting, I assume b=10 times hbar Plot Sin 10* p-1 ^4 10* p-1 ^4, {p, -3, 3}, AxesLabel→ {"p/p0", "Prob / (12/π)"}, PlotRange→
Page 6: tt-½ l :pc.,µ,.,,v, · 2018. 12. 10. · Q4(c), For plotting, I assume b=10 times hbar Plot Sin 10* p-1 ^4 10* p-1 ^4, {p, -3, 3}, AxesLabel→ {"p/p0", "Prob / (12/π)"}, PlotRange→
Page 7: tt-½ l :pc.,µ,.,,v, · 2018. 12. 10. · Q4(c), For plotting, I assume b=10 times hbar Plot Sin 10* p-1 ^4 10* p-1 ^4, {p, -3, 3}, AxesLabel→ {"p/p0", "Prob / (12/π)"}, PlotRange→
Page 8: tt-½ l :pc.,µ,.,,v, · 2018. 12. 10. · Q4(c), For plotting, I assume b=10 times hbar Plot Sin 10* p-1 ^4 10* p-1 ^4, {p, -3, 3}, AxesLabel→ {"p/p0", "Prob / (12/π)"}, PlotRange→
Page 9: tt-½ l :pc.,µ,.,,v, · 2018. 12. 10. · Q4(c), For plotting, I assume b=10 times hbar Plot Sin 10* p-1 ^4 10* p-1 ^4, {p, -3, 3}, AxesLabel→ {"p/p0", "Prob / (12/π)"}, PlotRange→
Page 10: tt-½ l :pc.,µ,.,,v, · 2018. 12. 10. · Q4(c), For plotting, I assume b=10 times hbar Plot Sin 10* p-1 ^4 10* p-1 ^4, {p, -3, 3}, AxesLabel→ {"p/p0", "Prob / (12/π)"}, PlotRange→
Page 11: tt-½ l :pc.,µ,.,,v, · 2018. 12. 10. · Q4(c), For plotting, I assume b=10 times hbar Plot Sin 10* p-1 ^4 10* p-1 ^4, {p, -3, 3}, AxesLabel→ {"p/p0", "Prob / (12/π)"}, PlotRange→
Page 12: tt-½ l :pc.,µ,.,,v, · 2018. 12. 10. · Q4(c), For plotting, I assume b=10 times hbar Plot Sin 10* p-1 ^4 10* p-1 ^4, {p, -3, 3}, AxesLabel→ {"p/p0", "Prob / (12/π)"}, PlotRange→
Page 13: tt-½ l :pc.,µ,.,,v, · 2018. 12. 10. · Q4(c), For plotting, I assume b=10 times hbar Plot Sin 10* p-1 ^4 10* p-1 ^4, {p, -3, 3}, AxesLabel→ {"p/p0", "Prob / (12/π)"}, PlotRange→
Page 14: tt-½ l :pc.,µ,.,,v, · 2018. 12. 10. · Q4(c), For plotting, I assume b=10 times hbar Plot Sin 10* p-1 ^4 10* p-1 ^4, {p, -3, 3}, AxesLabel→ {"p/p0", "Prob / (12/π)"}, PlotRange→
Page 15: tt-½ l :pc.,µ,.,,v, · 2018. 12. 10. · Q4(c), For plotting, I assume b=10 times hbar Plot Sin 10* p-1 ^4 10* p-1 ^4, {p, -3, 3}, AxesLabel→ {"p/p0", "Prob / (12/π)"}, PlotRange→
Page 16: tt-½ l :pc.,µ,.,,v, · 2018. 12. 10. · Q4(c), For plotting, I assume b=10 times hbar Plot Sin 10* p-1 ^4 10* p-1 ^4, {p, -3, 3}, AxesLabel→ {"p/p0", "Prob / (12/π)"}, PlotRange→
Page 17: tt-½ l :pc.,µ,.,,v, · 2018. 12. 10. · Q4(c), For plotting, I assume b=10 times hbar Plot Sin 10* p-1 ^4 10* p-1 ^4, {p, -3, 3}, AxesLabel→ {"p/p0", "Prob / (12/π)"}, PlotRange→
Page 18: tt-½ l :pc.,µ,.,,v, · 2018. 12. 10. · Q4(c), For plotting, I assume b=10 times hbar Plot Sin 10* p-1 ^4 10* p-1 ^4, {p, -3, 3}, AxesLabel→ {"p/p0", "Prob / (12/π)"}, PlotRange→
Page 19: tt-½ l :pc.,µ,.,,v, · 2018. 12. 10. · Q4(c), For plotting, I assume b=10 times hbar Plot Sin 10* p-1 ^4 10* p-1 ^4, {p, -3, 3}, AxesLabel→ {"p/p0", "Prob / (12/π)"}, PlotRange→
Page 20: tt-½ l :pc.,µ,.,,v, · 2018. 12. 10. · Q4(c), For plotting, I assume b=10 times hbar Plot Sin 10* p-1 ^4 10* p-1 ^4, {p, -3, 3}, AxesLabel→ {"p/p0", "Prob / (12/π)"}, PlotRange→
Page 21: tt-½ l :pc.,µ,.,,v, · 2018. 12. 10. · Q4(c), For plotting, I assume b=10 times hbar Plot Sin 10* p-1 ^4 10* p-1 ^4, {p, -3, 3}, AxesLabel→ {"p/p0", "Prob / (12/π)"}, PlotRange→
Page 22: tt-½ l :pc.,µ,.,,v, · 2018. 12. 10. · Q4(c), For plotting, I assume b=10 times hbar Plot Sin 10* p-1 ^4 10* p-1 ^4, {p, -3, 3}, AxesLabel→ {"p/p0", "Prob / (12/π)"}, PlotRange→
Page 23: tt-½ l :pc.,µ,.,,v, · 2018. 12. 10. · Q4(c), For plotting, I assume b=10 times hbar Plot Sin 10* p-1 ^4 10* p-1 ^4, {p, -3, 3}, AxesLabel→ {"p/p0", "Prob / (12/π)"}, PlotRange→
Page 24: tt-½ l :pc.,µ,.,,v, · 2018. 12. 10. · Q4(c), For plotting, I assume b=10 times hbar Plot Sin 10* p-1 ^4 10* p-1 ^4, {p, -3, 3}, AxesLabel→ {"p/p0", "Prob / (12/π)"}, PlotRange→
Page 25: tt-½ l :pc.,µ,.,,v, · 2018. 12. 10. · Q4(c), For plotting, I assume b=10 times hbar Plot Sin 10* p-1 ^4 10* p-1 ^4, {p, -3, 3}, AxesLabel→ {"p/p0", "Prob / (12/π)"}, PlotRange→
Page 26: tt-½ l :pc.,µ,.,,v, · 2018. 12. 10. · Q4(c), For plotting, I assume b=10 times hbar Plot Sin 10* p-1 ^4 10* p-1 ^4, {p, -3, 3}, AxesLabel→ {"p/p0", "Prob / (12/π)"}, PlotRange→
Page 27: tt-½ l :pc.,µ,.,,v, · 2018. 12. 10. · Q4(c), For plotting, I assume b=10 times hbar Plot Sin 10* p-1 ^4 10* p-1 ^4, {p, -3, 3}, AxesLabel→ {"p/p0", "Prob / (12/π)"}, PlotRange→
Page 28: tt-½ l :pc.,µ,.,,v, · 2018. 12. 10. · Q4(c), For plotting, I assume b=10 times hbar Plot Sin 10* p-1 ^4 10* p-1 ^4, {p, -3, 3}, AxesLabel→ {"p/p0", "Prob / (12/π)"}, PlotRange→
Page 29: tt-½ l :pc.,µ,.,,v, · 2018. 12. 10. · Q4(c), For plotting, I assume b=10 times hbar Plot Sin 10* p-1 ^4 10* p-1 ^4, {p, -3, 3}, AxesLabel→ {"p/p0", "Prob / (12/π)"}, PlotRange→
Page 30: tt-½ l :pc.,µ,.,,v, · 2018. 12. 10. · Q4(c), For plotting, I assume b=10 times hbar Plot Sin 10* p-1 ^4 10* p-1 ^4, {p, -3, 3}, AxesLabel→ {"p/p0", "Prob / (12/π)"}, PlotRange→
Page 31: tt-½ l :pc.,µ,.,,v, · 2018. 12. 10. · Q4(c), For plotting, I assume b=10 times hbar Plot Sin 10* p-1 ^4 10* p-1 ^4, {p, -3, 3}, AxesLabel→ {"p/p0", "Prob / (12/π)"}, PlotRange→
Page 32: tt-½ l :pc.,µ,.,,v, · 2018. 12. 10. · Q4(c), For plotting, I assume b=10 times hbar Plot Sin 10* p-1 ^4 10* p-1 ^4, {p, -3, 3}, AxesLabel→ {"p/p0", "Prob / (12/π)"}, PlotRange→
Page 33: tt-½ l :pc.,µ,.,,v, · 2018. 12. 10. · Q4(c), For plotting, I assume b=10 times hbar Plot Sin 10* p-1 ^4 10* p-1 ^4, {p, -3, 3}, AxesLabel→ {"p/p0", "Prob / (12/π)"}, PlotRange→
Page 34: tt-½ l :pc.,µ,.,,v, · 2018. 12. 10. · Q4(c), For plotting, I assume b=10 times hbar Plot Sin 10* p-1 ^4 10* p-1 ^4, {p, -3, 3}, AxesLabel→ {"p/p0", "Prob / (12/π)"}, PlotRange→
Page 35: tt-½ l :pc.,µ,.,,v, · 2018. 12. 10. · Q4(c), For plotting, I assume b=10 times hbar Plot Sin 10* p-1 ^4 10* p-1 ^4, {p, -3, 3}, AxesLabel→ {"p/p0", "Prob / (12/π)"}, PlotRange→

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]

Page 36: tt-½ l :pc.,µ,.,,v, · 2018. 12. 10. · Q4(c), For plotting, I assume b=10 times hbar Plot Sin 10* p-1 ^4 10* p-1 ^4, {p, -3, 3}, AxesLabel→ {"p/p0", "Prob / (12/π)"}, PlotRange→

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

Page 37: tt-½ l :pc.,µ,.,,v, · 2018. 12. 10. · Q4(c), For plotting, I assume b=10 times hbar Plot Sin 10* p-1 ^4 10* p-1 ^4, {p, -3, 3}, AxesLabel→ {"p/p0", "Prob / (12/π)"}, PlotRange→

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

Page 38: tt-½ l :pc.,µ,.,,v, · 2018. 12. 10. · Q4(c), For plotting, I assume b=10 times hbar Plot Sin 10* p-1 ^4 10* p-1 ^4, {p, -3, 3}, AxesLabel→ {"p/p0", "Prob / (12/π)"}, PlotRange→

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

Page 39: tt-½ l :pc.,µ,.,,v, · 2018. 12. 10. · Q4(c), For plotting, I assume b=10 times hbar Plot Sin 10* p-1 ^4 10* p-1 ^4, {p, -3, 3}, AxesLabel→ {"p/p0", "Prob / (12/π)"}, PlotRange→

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

Page 40: tt-½ l :pc.,µ,.,,v, · 2018. 12. 10. · Q4(c), For plotting, I assume b=10 times hbar Plot Sin 10* p-1 ^4 10* p-1 ^4, {p, -3, 3}, AxesLabel→ {"p/p0", "Prob / (12/π)"}, PlotRange→

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

Page 41: tt-½ l :pc.,µ,.,,v, · 2018. 12. 10. · Q4(c), For plotting, I assume b=10 times hbar Plot Sin 10* p-1 ^4 10* p-1 ^4, {p, -3, 3}, AxesLabel→ {"p/p0", "Prob / (12/π)"}, PlotRange→

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

Page 42: tt-½ l :pc.,µ,.,,v, · 2018. 12. 10. · Q4(c), For plotting, I assume b=10 times hbar Plot Sin 10* p-1 ^4 10* p-1 ^4, {p, -3, 3}, AxesLabel→ {"p/p0", "Prob / (12/π)"}, PlotRange→

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