QuantumPhysics.dvi

In the Hamiltonian formalism, the momentum Πicanonically conjugate toAiis given by Πi= ∂L ∂A ̇i =ε 0 A ̇i=−ε 0 Ei (19.14) Now, i ...

whose general solution is a linear combination ofeiωt ande−iωt. Combining now the solution for all equations onC, we have C(t,k) ...

To evaluateHin terms of the vector potential, in transverse gauge, we usethe expressionsE= −∂tAandB=∇×A, which themselves may be ...

so that we have the following expression for the full Hamiltonian, H= ∫ d (^3) k (2π)^3 ∑ α |k|aα(k)∗αα(k) (19.36) The Poisson b ...

19.4 Momentum in terms of radiation oscillators The classical electro-magnetic field carries momentum, which may be expressed in ...

The only novelty here, as compared with systems of finite numbers of harmonic oscillators, is that here, the oscillators are ind ...

It will be assumed throughout that this state is unique and normalized as follows, ‖|∅〉λ=〈∅|∅〉= 1 (19.55) As an immediate result ...

whose energy and momentum is readily computed, and we have, H|ψ〉=E|ψ〉 E=|k 1 |+···+|kn| P|ψ〉=p|ψ〉 p=k 1 +···+kn (19.62) Energy a ...

19.7 Bose-Einstein and Fermi-Dirac statistics In classical mechanics, particles aredistinguishable, namely each particle may be ...

First of all, a free fermion may be expected to be equivalent to a collection of oscillators, just as bosonic free fields were. ...

Using the fact that creation operators always anti-commutewith one another, it is straightforward to show that b†σ 1 (~k 1 )···b ...

ExpressingCin terms of the oscillator modes, we obtaint-dependent terms andt-independent terms. The former vanish upon integrati ...

19.9 The Casimir Effect on parallel plates Thus far, we have dealt with the quantization of the free electromagnetic fieldsE~ an ...

Fig18(b). Calculation of the frequencies There are three distinct regimes in which the frequencies ofthe quantum oscillators ne ...

Next, we are interested in taking the size of the boxLvery large compared to all other scales in the problem. The levels innxand ...

where the Bernoulli numbers are defined by x ex− 1 = ∑∞ n=0 Bn xn n! B 2 =− 1 6 , B 4 =− 1 30 ,··· (19.99) Assuming a sharp cuto ...

20 Photon Emission and Absorption In this chapter, we couple the quantized electro-magnetic field to quantized matter, such as e ...

obtained by simply promoting the classicalAfield to a quantum operator. The Hilbert spaceHof the full system is the tensor produ ...

course, in cases where the orbital term gives vanishing contribution, the electron magnetic moment term may become important, an ...

As a result, the productk·x∼αis small compare to 1, and the exponential may be approximated by 1, at the same order of approxima ...