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
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