Wavepackets etc.

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ψk (x) =1

2πe ikx, Ek =

h2k 2

2m

Plane wave eigenstates:

• scattering states• non-normalizable• continuous spectra

x

Free particle

Wavepackets

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Φ(k) = dx−∞

∞

∫ ψ k*(x)Ψ(x,0)with

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1 = dk−∞

∞

∫ Φ(k)2Normalization:

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Ψ(x, t) = dk−∞

∞

∫ Φ(k)ψ k (x)e−iωk t , ωk =hk 2

2m

Linear superpositions of plane wave solutions

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Ψ(x,0) =0, |x|>a

12a

, |x|<a

{

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Φ(k) =1

2πdx

1

2ae−ikx =

1

πa−a

a

∫ sin ka

k

Example 1

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Ψ(x, t) =1

4πadk

−∞

∞

∫ sinka

ke ikxe−iωk t , ωk =

hk 2

2m

Unfortunately, the integrand isbadly behaved and difficult to evaluate even numerically.

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Ψ(x, t) = dk−∞

∞

∫ Φ(k)ψ k (x)e−iωk t , Φ(k) = dy−∞

∞

∫ ψ k*(y)Ψ(y,0)

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Ψ(x, t) = dy−∞

∞

∫ Ψ(y,0) dk−∞

∞

∫ ψ k*(y)ψ k (x)e−iωk t = dy

−∞

∞

∫ G(x,y;t)Ψ(y,0)

Propagator

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G(x,y;t) = dk−∞

∞

∫ ψ k*(y)ψ k (x)e−iωk t

=1

2πdk

−∞

∞

∫ e−ik(x−y )e−τk 2

=1

2τe−(x−y )2 / 4τ

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G(x,y;t) is a free particle propagator

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τ = ih

2mt “imaginary time”

Calculate evolution of the wavefunction using the propagator

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Ψ(x, t) = dy−∞

∞

∫ G(x, y;t)Ψ(y,0)

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G(x, y;t) =1

2τe−(x−y )2 / 4τ

QuickTime™ and aGIF decompressor

are needed to see this picture.

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Ψ(x,0) =2a

π

⎛

⎝ ⎜

⎞

⎠ ⎟

1

4e−ax 2

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Φ(k) =1

(2πa)1/ 4e−k 2 / 4 a

Example 2: Gaussian wavepacket

QuickTime™ and aGIF decompressor

are needed to see this picture.

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Ψ(x, t) =2a

π

⎛

⎝ ⎜

⎞

⎠ ⎟

1

4 exp[−ax 2 /(1+ 2ihat /m)]

1+ 2ihat /m

Example 3: moving Gaussian wavepacket

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Ψ(x,0) =2a

π

⎛

⎝ ⎜

⎞

⎠ ⎟

1

4e−ax 2

e ilxHow do we know this is a moving wavepacket?

Calculate the momentumexpectation value:

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ˆ p = dx∫ Ψ*(−ih∂x )Ψ = hl

QuickTime™ and aGIF decompressor

are needed to see this picture.

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Ψ(x, t) =2a

π

⎛

⎝ ⎜

⎞

⎠ ⎟

1

4e−l 2 / 4 a exp[−a(x − il /2a)2 /(1+ 2ihat /m)]

1+ 2ihat /m€

Re[Ψ(x, t)]

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Ψ(x, t)

Example 4: wavepacket reflected from a hard wall

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Ψ(0, t) = 0

Boundary condition:

Eigenstates:

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ψk (x) = Aθ(−x)sin kx

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G(x, y;t) = dk−∞

∞

∫ ψ k*(y)ψ k (x)e−iωk t

=θ(−x)θ(−y)

2τ[e−(x−y )2 / 4τ − e−(x +y )2 / 4τ ]

Propagator: 0 x0

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V = 0

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V = ∞

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Ψ(x, t) = dy−∞

∞

∫ G(x, y;t)Ψ(y,0)

QuickTime™ and aGIF decompressor

are needed to see this picture.

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Re[Ψ(x, t)]

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Ψ(x, t)

Wavepackets reflected from steps & barriers

•For very nice and instructive animations visit http://www.quantum-physics.polytechnique.fr/en/pages/p0103.html