QuantumPhysics.dvi

The norms ofJ±|j,m±〉were already computed earlier, and thus require that λ(j)−m+(m++ 1) = 0 λ(j)−m−(m−−1) = 0 (5.97) Eliminating ...

The operatorL^2 may be expressed solely in terms of the coordinatesθandφ, but we shall not need its explicit form here. Instead, ...

For an electron we haveq 1 =−e, and for a nucleus of withZprotons we haveq 2 =Ze. In addition, in cgs units (which is what we ha ...

This equation is of a well-known form, namely that of a confluent hypergeometric function, or Kummer function, gℓ,ε(x) =M ( ℓ+ 1 ...

The operatorH 1 will be self-adjoint if a domainD(H 1 ) can be chosen for the functionsψ andφsuch thatj(0) = 0 forany pairψ,φ∈ D ...

Before launching into any math, let’s solve the Schr ̈odinger equation forH 2 , i ̄h ∂ ∂t ψ(t,x) =i ̄hc ∂ ∂x ψ(t,x) (5.126) or e ...

5.9.3 Example 3: One-dimensional Dirac-like operator in a box A lesson we have learned from Example 2 is that the momentum opera ...

6 Quantum Mechanics Systems Quantum systems associated with systems of classical mechanics are fundamental. In fact, it is for t ...

The variation is then computed using standard chain rule, δS[q] = S[q+δq]−S[q] = ∫t 2 t 1 dt ( L(q+δq,q ̇+δq ̇;t)−L(q,q ̇;t) ) = ...

6.2 Hamiltonian mechanics Returning to the general case, with LagrangianL(q,q ̇;t), one defines the momentum pi canonically conj ...

Since the number of dynamical variables on phase space has been doubled up, Hamilton’s equations are now first order. There is a ...

formulation. A second is via thefunctional or path integralwhich directly produces prob- ability amplitudes from a classical mec ...

have a unique quantum Hamiltonian, given by the correspondence principle, H(Q,P) = P^2 2 m +V(Q) (6.32) In the position represen ...

7 Charged particle in an electro-magnetic field Of fundamental importance is the problem of an electrically chargedparticle in t ...

Under gauge transformations, A→A′=A+∇θ Φ→Φ′= Φ− ∂θ ∂t (7.7) the fieldsBandEare invariant (i.e. B→B′ =BandE→E′ =E) for an arbitra ...

7.2 Constant Magnetic fields An important special case is when the electric and magnetic fields areconstant in time, and uniform ...

and their adjointsa† 1 anda† 2. By construction they obey canonical commutation relations [ai,a†j] =δij fori,j= 1,2, while [a 1 ...

7.3 Landau Levels For a charged particle in a magnetic field without the harmonic oscillator present, we have ω= 0 andω−= 0, and ...

Landau level satisfy the differential equation ( ∂ ∂z ̄ + mωB ̄h z ) ψ(0)(z, ̄z) = 0 (7.33) Its general solution is straightforw ...

solenoid, the gauge potentialAmust satisfy∇×A= 0. Nonetheless, the gauge potential cannot vanish outside, because we have in vie ...