continuity equation for probability density

continuity equation for probability density

probability-density current

time-dependent Schrödinger equation

i~��(⇥r , t)�t

=

✓�~22m⇥r2 + V (⇥r , t)

◆�(⇥r , t)

Born interpretation: |Ψ(r,t)|2 is probability density for finding particle at time t at position r

�|�|2

�t= �⇥⇥ ·⇥j

�j = �i~2m

�� �r��� �r�

�

Crank-Nicolson algorithm

✓1 +1

2iH�t

◆⇥(tn+1) =

✓1�1

2iH�t

◆⇥(tn)

e�iH�t ⇡1� 12 iH�t1 + 12 iH�t

separation of variables

time-independent potential

A(t) = A0 e�iEt/~

general solution: linear combination of eigenstates

i~��(⇥r , t)�t

=

✓�~22m⇥r2 + V (⇥r)

◆�(⇥r , t)

time-independent Schrödinger equation (eigenvalue problem)

ansatz: (~r , t) = A(t)'(~r)

i~@A(t)@t

'(~r) = A(t)E '(~r) = A(t)

✓�~22m~r2 + V (~r)

◆'(~r)

✓�~22m~r2 + V (~r)

◆'(~r) = E '(~r)

(~r , t) =X

n

an e�iEnt/~ 'n(~r)

particle in a box

discrete energies zero-point energy

increasing number of nodes

symmetry of potential symmetry of solutions (density)

even/odd eigenfunctions

En =~22m

✓n⇡

Lz

◆2boundary conditions ⇒ quantization

⇥n(z) =

r2

Lzsin

✓n� z

Lz

◆

particle in a box

En =~22m

✓n⇡

Lz

◆2boundary conditions ⇒ quantization

⇥n(z) =

r2

Lzsin

✓n� z

Lz

◆

free wave packets

free time-independent Schrödinger equation: �~22m

d2

dx2�(x) = E�(x)

eigenfunctions and -values: �k(x) = C eikx ; Ek =

~2k22m

normalization:Zdx |�k(x)|2 = C2

Z 1

�1dx =??

improper wave functions:

with ‘normalization’ as in Fourier transform:Zdx ⇤k 0(x)⇤k(x) =

1

2⇥

Z 1

�1dx e i(k�k

0)x = �(k � k 0)

⇥k(x) =1p2�e ikx

wave packet: normalizable linear combination of plane waves

�(x, t) =1p2�

Zdk ⇥̃(k) e i(kx��(k) t)

Zdk |�̃(k)|2 = 1with

approximation to Dirac delta function

-20

-10

0

10

20

30

40

50

-2 -1 0 1 2

k-k’

L=10

L=20

L=50

sin(Lx)

x

Z 1

�1dz e i(k�k

0)z = limL!1

2sin((k � k 0)L/2)(k � k 0) = 2⇥ �(k � k 0)

Z L/2

�L/2dz e i(kn�km)z =

2sin((kn � km)L/2)kn � km

= L �n,m (kn = 2⇡n/L)<latexit sha1_base64="0CjoeavBCD/2nTUrFDGvHVauKiw=">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</latexit><latexit sha1_base64="0CjoeavBCD/2nTUrFDGvHVauKiw=">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</latexit><latexit sha1_base64="0CjoeavBCD/2nTUrFDGvHVauKiw=">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</latexit><latexit sha1_base64="0CjoeavBCD/2nTUrFDGvHVauKiw=">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</latexit>

Gaussian wave packet

⇤̃(k) =1pp2� ⇥k

e�(k�k0)2/4�2k

probability density for finding momentum ħk is Gaussian of width σk, centered at k0

|�̃(k)|2

probability density for finding particle at time t in position x is a Gaussianof width �x

p1 + ⇥2�t

2, where �x = 1/2�k and ⇥� = ~/2m�2x ,with its center moving with the group velocity vg = ~k0/m.

�(x, t) =1p

(2�)3/2⇥k

Zdk e�(k�k0)

2/4�2k e i(kx�⇥(k)t)

=1qp

2�⇥x(1 + i⇤�t)e i(k0x�⇥(k0)t) e

� (x�vg t)2

4�2x (1+i⇥� t)

spreading of wave packet

0

1

2

3

4

5

-4 -2 0 2 4

�x(

t)/�

x

�� t

Heisenberg limit: σx σk=½

spre

ading

of en

semble

of cl

assic

al pa

rticles

with G

auss

ian m

omen

tum di

stribu

tion

wave packet

0

0.5

1

1.5

2

2.5

3

-20 -15 -10 -5 0 5 10 15 20

t=0.00

0

0.5

1

1.5

2

2.5

3

-20 -15 -10 -5 0 5 10 15 20

t=1.00

0

0.5

1

1.5

2

2.5

3

-20 -15 -10 -5 0 5 10 15 20

t=2.00

0

0.5

1

1.5

2

2.5

3

-20 -15 -10 -5 0 5 10 15 20

t=3.00

superposition of two wave packets

0

0.5

1

1.5

2

2.5

3

-20 -15 -10 -5 0 5 10 15 20

t=1.00

0

0.5

1

1.5

2

2.5

3

-20 -15 -10 -5 0 5 10 15 20

t=2.00

0

0.5

1

1.5

2

2.5

3

-20 -15 -10 -5 0 5 10 15 20

t=3.00

0

0.5

1

1.5

2

2.5

3

-20 -15 -10 -5 0 5 10 15 20

t=4.10

wave packet hitting infinite wall

0

0.5

1

1.5

2

2.5

3

-20 -15 -10 -5 0 5 10 15 20

t=1.00

0

0.5

1

1.5

2

2.5

3

-20 -15 -10 -5 0 5 10 15 20

t=1.50

0

0.5

1

1.5

2

2.5

3

-20 -15 -10 -5 0 5 10 15 20

t=2.00

0

0.5

1

1.5

2

2.5

3

-20 -15 -10 -5 0 5 10 15 20

t=3.00