continuity equation for probability densitycontinuity equation for probability density continuity...

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continuity equation for probability density continuity equation for probability density probability-density current time-dependent Schrödinger equation i ~ Ψ( r,t ) t = - ~ 2 2m r 2 + V ( r,t ) Ψ( r,t ) Born interpretation: |Ψ(r,t)| 2 is probability density for finding particle at time t at position r |Ψ| 2 t = - · j j = - i ~ 2m ( Ψ rΨ - Ψ r Ψ )

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Transcript of continuity equation for probability densitycontinuity equation for probability density continuity...

  • 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)

    [email protected](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 e ikx ; 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)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

  • 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

    semb

    le of

    class

    ical p

    artic

    les

    with

    Gaus

    sian 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