Harmonic oscillator

Newton's law: ma Kx

dL  L 0 Euler - Lagrange equations:    dt qi  q i

Lagrangian 2 2 mx Kx (constructed by L x,x    inspection) 2 2 L p  mx Conjugate variable: x p2 Kx 2 H px  L  Legendre transformation: 2m 2

 2d 2 Kx 2 Quantize: p  i H   x 2m dx2 2 LC circuit

dI dV Classical equations VL IC  Q CV dt dt

Q dQ2 L C dt 2

d L L    0 Euler - Lagrange equation: dt Q  Q

 2 2 Lagrangian  LQ Q L Q,Q   (constructed by   2 2C inspection) LC circuit

  pL  LQ Conjugate variable: Q

 2 2  2 LQ Q Legendre transformation: H LQ   L 2 2C

 Quantize: p  i Q

2d 2 Q 2 H   E 2L dQ2 2 C Transmission line

iCZ2  2 LCZ  iL    0

dV dI dI dV L inductance/m  L  C C capacitance/m dx dt dx dt

dV2 dV 2  LC dx2 dt 2

Vk  V0 exp ikx  t normal mode solution: Ik  I0 exp ikx  t   Each normal mode moves independently from the other normal modes Transmission line

iCZ2  2 LCZ  iL    0

Substituting the normal mode solution VV0 exp ikx  t

dV2 dV 2 into the wave equation  LC k2  LC 2 dx2 dt 2 k yields the dispersion relation    ck LC C V L I V Z   L I C

An infinite transmission line is resistive, typically ~ 50 W. Transmission line

iCZ2  2 LCZ  iL    0

Wave equation

2 2 2 dV dV 1 c  c  dx2 dt 2 LC c is the speed of waves

Not clear what mass we should use in the Schrödinger equation The Schrödinger equation is for amateurs

 Lagrangian L  x,x (constructed by Euler - Lagrange inspection) equations: d   LL  0 dt x  x  Conjugate p  L x variable: Classical equations of motion (Newton's law)

 Legendre transformation: H px L

 Quantize: p  i x Transmission line

iCZ2  2 LCZ  iL    0

2 2 2 dV dV c  dx2 dt 2

normal mode solution: Vk  V0 exp ikx  t

2 2 2 d Vk ckVk  dt 2

Each normal mode moves independently from the other normal modes Lagrangian

Construct the Lagrangian 'by inspection'. The Euler-Lagrange equation and the classical equation of motion for a normal mode are,

 L  L 2   0. 2 2  Vk   ckVk  . tVk   V k t 2 classical equation for the mode k

 2 2 2 Vk c k 2 The Lagrangian is, L   V 2 2 k Hamiltonian

 2 2 2 V c k 2 L k  V 2 2 k

The conjugate variable to Vk is,  L    Vk Vk The Hamiltonian is constructed by performing a Legendre transformation,  2 2 2   Vk c k 2 HVVkk L  V k 2 2  To quantize we replace the conjugate variable by i Vk 2 2 2 2  d ck 2 2 Vk   E  2dVk 2 Quantum solutions

2 2 2 2 2 2  d K 2  d ck 2 2 x  E  2 Vk   E  2m dx 2 2dVk 2

This equation is mathematically equivalent to the harmonic oscillator.

 1 j  0,1, 2, E  j  2  spring constant K     c2 k 2 m mass - spring wave mode    c k

j is the number of photons. Dissipation in Quantum mechanics

Transmission line

=

C I V L V L Z   I C

An infinite transmission line is resistive Nitrogen

 iE0 t   0 exp   

 iEt1  1 exp    Dissipation in quantum mechanics

Quantum coherence is maintained until the decoherence time. This depends on the strength of the coupling of the quantum system to other degrees of freedom.

Decay is the decoherence time. Dissipation in solids

At zero electric field, the electron eigen states are Bloch states. Each Bloch state has a k vector. The average value of k = 0 (no current). At finite electric field, the Bloch states are no longer eigen states but we can calculate the transitions between Bloch states using Fermi's golden rule. The final state may include an electron state plus a phonon. The average value of k is not zero (finite current).

The phonons carry the energy away like a transmission line. Institute of Solid State Physics Technische Universität Graz

The quantization of the electromagnetic field

Wave nature and the particle nature of light Unification of the laws for electricity and magnetism (described by Maxwell's equations) and light Quantization of fields Derive the Bose-Einstein function Planck's radiation law Serves as a template for the quantization of noninteracting bosons: phonons, magnons, plasmons, and other quantum particles that inhabit solids.

http://lamp.tu-graz.ac.at/~hadley/ss2/emfield/quantization_em.php Maxwell's equations

  E    0 B  0   B E  t    E B 0 J   0 0 t The vector potential

  B A   A E V  t  Maxwell's equations in terms of A Coulomb gauge A  0  A   0  A 0 t  t A 0  Vector identity  A      A   AA tt  tt   2 A A 0  0 t 2 The wave equation    2 A A 0  0 t 2    2 Using the identity A (), A  A   2 2 2  A wave equation c A  . t 2

    normal mode solutions have the form: Art(,) A exp(( ikr   t ))

Substituting the normal mode solution in the wave equation results in the dispersion relation    c k EM waves propagating in the x direction

 AA0 cos( kxx   tz ) ˆ

The electric and magnetic fields are   A E Akxtz0 sin(x   )ˆ ,  t B AkAx0 sin( kx x   ty )ˆ . Lagrangian

To quantize the wave equation we first construct the Lagrangian 'by inspection'. The Euler-Lagrange equation and the classical equation of motion are,

L   L 2   0. 2 2  As   ck As  . tAs   A s t 2 classical equation for the normal mode k  2 2 2 A c k 2 The Lagrangian is, Ls  A 2 2 s Hamiltonian

 2 2 2 A c k 2 Ls  A 2 2 s

The conjugate variable to As is, L    As As The Hamiltonian is constructed by performing a Legendre transformation,  2 2 2   As c k 2 HAAL   A ss2 2 s  To quantize we replace the conjugate variable by i As 2 2 2 2  d ck 2 2 As  E  2dAs 2 Quantum solutions

2 2 2 2  d ck 2 2 As  E  2dAs 2

This equation is mathematically equivalent to the harmonic oscillator.

 1 j  0,1, 2, Es ss j  2  s 

s c k s

js is the number of photons in mode s. Institute of Solid State Physics Technische Universität Graz

Non-interacting boson systems

Photons, phonons, magnons, plasmons can be approximated as non-interacting bosons. To calculate their thermodynamic properties:

Construct the partition function

Nq N q  Eq  EN q q   Zgr (,) T   exp exp   exp   qkTB kT B q  kT B 

kB T   ln(Z gr ) V Deduce the thermodynamic properties:   n  f  n  s   T