Inclusion of the electromagnetic field in Quantum …physics.wustl.edu/wimd/524EMa.pdfE&M Inclusion...

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E&M Inclusion of the electromagnetic field in Quantum Mechanics similar to Classical Mechanics but with interesting consequences Maxwell’s equations Scalar and vector potentials Lorentz force Transform to Lagrangian Then Hamiltonian Minimal coupling to charged particles

Transcript of Inclusion of the electromagnetic field in Quantum …physics.wustl.edu/wimd/524EMa.pdfE&M Inclusion...

E&M

Inclusion of the electromagnetic field in Quantum Mechanics

similar to Classical Mechanics but with interesting consequences

• Maxwell’s equations• Scalar and vector potentials

• Lorentz force

• Transform to Lagrangian

• Then Hamiltonian

• Minimal coupling to charged particles

∇ ·E(x, t) = 4πρ(x, t)

∇ ·B(x, t) = 0

∇×E(x, t) = −1

c

∂tB(x, t)

∇×B(x, t) =1

c

∂tE(x, t) +

cj(x, t)

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Maxwell’s equations

Gaussian units

E = −∇Φ− 1

c

∂A

∂t

B = ∇×A

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Scalar and Vector potentialQuantum applications require replacingelectric and magnetic fields! implies

From Faraday

so

or

in terms of vector and scalar potentials.Homogeneous equations are automatically solved.

∇ ·B = 0

∇×�E +

1

c

∂tA

�= 0

E +1

c

∂tA = −∇Φ

∇ ·A+1

c

∂Φ

∂t= 0

∇2Φ+1

c

∂t(∇ ·A) = −4πρ

∇2A− 1

c2∂2A

∂t2−∇

�∇ ·A+

1

c

∂Φ

∂t

�= −4π

cj

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Gauge freedom

Remaining equations using

To decouple one could choose (gauge freedom)

more later… first

∇× (∇×A) = ∇ (∇ ·A)−∇2A

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Coupling to charged particles

Lorentz

Rewrite

Note

and

So that withF = −∇U +d

dt

∂U

∂vU = qΦ− q

cA · v

∂A

∂t+ (v ·∇)A =

dA

dt

F = q

�E +

1

cv ×B

v × (∇×A) = ∇ (v ·A)− (v ·∇)A

F = q

�−∇Φ− 1

c

∂A

∂t+

1

cv × (∇×A)

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CheckYields Lorentz from

Equations of motion

Generalized momentum

Solve for v and substitute in Hamiltonian--> Hamiltonian for a charged particle

H = p · v − L =

�p− q

cA�2

2m+ qΦ

p =∂L

∂v= mv +

q

cA

d

dt

∂L

∂v− ∂L

∂x= 0

L = T − U =1

2mv2 − qΦ+

q

cA · v

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Include external electromagnetic field in QM• Static electric field: nothing new (position --> operator)• Include static magnetic field with momentum and position

operators

• Note velocity operator

• Note Hamiltonian not “free” particle one

• Use

• to show that

• Gauge independent! So think in terms of

v =1

m

�p− q

cA�

H =

�p− q

cA(x)�2

2m

[pi, Aj ] =�i

∂Aj

∂xi

[vi, vj ] = iq�m2c

�ijkBk

H =1

2m|v|2

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Include external electromagnetic field• Include uniform magnetic field• For example by

• Only nonvanishing commutator

• Write Hamiltonian as

• but now so this corresponds to free particle motion parallel to magnetic field (true classically too)

• Only consider

• Operators don’t commute but commutator is a complex number!

• So...

B(x) = Bz

[vx, vy] = iq�Bm2c

H =1

2m

�v2x + v

2y + v

2z

vz =pzm

H =1

2m

�v2x + v

2y

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Harmonic oscillator again...• Motion perpendicular to magnetic field --> harmonic oscillator• Introduce

• with cyclotron frequency

• Straightforward to check

• So Hamiltonian becomes

• and consequently spectrum is (called Landau levels)

ωc =qB

mc

�a, a†

�= 1

H = �ωc

�a†a+ 1

2

En = �ωc (n+ 12 ) n = 0, 1, ....

a =

�m

2�ωc(vx + ivy)

a† =

�m

2�ωc(vx − ivy)

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Aharanov-Bohm effect• Consider hollow cylindrical shell

• Magnetic field inside inner cylinder either on or off

• Charged particle confined between inner and outer radius as well as top and bottom

A =Br212r

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Discussion• Without field:

– Wave function vanishes at the radii of the cylinders as well as top and bottom --> discrete energies

• With field (think of solenoid)– No magnetic field where the particle moves inside in z-direction and

constant

– Spectrum changes because the vector potential is needed in the Hamitonian

– Use Stokes theorem

– Only z-component of magnetic field so left-hand side becomes

– for any circular loop outside inner cylinder (and centered)

– Vector potential in the direction of and line integral -->

– Resulting in modifying the Hamiltonian and the spectrum!!

S(∇×A) · n da =

CA · d�

S(∇×A) · n da =

SBθ(r1 − ρ)da = Bπr21

φ 2πr

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Example• No field• Example of radial wave function

• Problem solved in cylindrical coordinates

• Also with field -->