On the Heisenberg and Schrödinger Pictures

A quantum theory for a one-electron system can be developed in either Heisenberg picture or Schrödinger picture. For a many-electron system, a theory must be developed in the Heisenberg picture, and the indistinguishability and Pauli’s exclusion principle must be incorporated. The hydrogen atom energy levels are obtained by solving the Schrödinger energy eigenvalue equation, which is the most significant result obtained in the Schrödinger picture. Both boson and fermion field equations are nonlinear in the presence of a pair interaction.


Introduction
Both Heisenberg (HP) and Schrödinger pictures (SP) are used in quantum theory.Schrödinger solved Schrödinger eigenvalue equation for a hydrogen atom, and obtained the atomic energy levels.Heisenberg discussed the uncertainty principle based on the fundamental commutation relations.Both pictures are equivalent in dealing with a one-electron system.In dealing with many electrons or many photons a theory must be developed in the HP, incorporating the indistinguishability and Pauli's exclusion principle.A quantum theory must give a classical result in some limit.We will see that this limit is represented by 0 →  .The HP, and not the SP, give the correct results for a many-particle system.The quantum field equation is nonlinear if a pair interaction exists.

One Electron Systems
We consider an electron in a potential energy field ( ) V r , where r is a position vector.In the Cartesian representation ( ) ( ) , , , , .x y z q q q = ≡ r (1) The canonical momentum p is , , , , .
x y z p p p p p p = ≡ p (2) The Hamiltonian H of the system is In the HP the coordinates and momenta ( ) , q p are regarded as Hermitean operators satisfying the fundamental commutation relations (quantum conditions): where jk δ is Kronecker's delta , h ≡ Planck constant.The equations of motion for j q and j p are The two equations can be included in a single equation: where ξ represent any physical observable made out of the components of the position x y z l l l = × = l r p can be included also.Dirac has shown that in the small  limit: is the classical Poisson brackets.
In the SP we use the equivalence relations: and write down the Schrödinger wave equation as The function ψ is called the wave function.Normally, it is normalized such that ( ) ( ) where Ω is a normalization volume.The quantum average of an observable ( ) , ξ r p is defined by If we use Dirac's ket and bra notations, then we can see the theoretical structures more compactly [1].
Using Equations ( 14) and (15), we obtain This equation, called the quantum Liouville equation, has a reversed sign compared with the equation of motion for ξ , see Equation (7).We can express the quantum average ξ of an observable ξ as where tr denotes a one-particle trace.Operators under a trace commute.We assume that the Hamiltonian H in Equation ( 11) is a constant of motion.Then, Equation ( 14) can be reduced to the energy ( ) 2 after using a separation of variable method for solving Equation (11).Equation ( 20) is known as the Schrödinger energy-eigenvalue equation.The hydrogen atom energy-levels can be obtained from Equation (20) with ( ) Except for simple systems such as free electrons and simple harmonic oscillators, the Heisenberg equation of motion (7) [or the quantum Liouville Equation (18)] are harder to solve.This is so because the numbers of unknowns in the n n × matrix are more numerous than in the 1 n × vector.

Difficulties with the SP
The following items have difficulties in the SP.They cannot be addressed properly.
(a) The Classical Mechanical Limit ( ) 0 →  Dirac showed that the fundamental commutation relations (8) can also be applied to a many particle system only if the Cartesian coordinates and momenta are used.The equation of motion (7) in the HP can be reduced to the classical equation of motion: The Schrödinger equation of motion (11) does not have such a simple limit.(b) Indistinguishability All electrons are identical (indistinguishable) to each other.This is known as the indistinguishability.This property can be stated as follows: Consider a system of N electrons interacting with each other characterized by the Hamiltonian: where ( ) j ε is the kinetic energy and is the pair interaction energy.Here the upper indices j and k denote the electrons.The indistinguishability requires that [ ] where  are the permutation operators.For a three-particle system the permutation operators are The order of the permutation group for an N -particle system is !N .The total momentum, the total angular momentum, and the total mass satisfy the same Equation (24).We may express this by [ ] (c) Boson Creation and Annihilation Photons are bosons with full spin.They can be created and annihilated spontaneously.These processes can only be described by using creation and annihilation operators ( ) † , a a both of which move, following the Heisenberg equations of motion.One can no more limit the number of bosons in the system.
(d) The Second Quantization for Bosons Bosons can be treated using second-quantized operators ( ) † , a a satisfying the Bose commutation rules: † † † † † , , , , , 0, a a a a a a a a a a where ( ) , µ ν indicates particle states.Both operators a and † a move, following the Heisenberg equations of motion, e.g.
where H is a many-boson Hamiltonian.The Hamiltonian for free photons is given by † , Both operators c and † c move, following the Heisenberg equations of motion.If the system contains many electrons, then we must consider Pauli's exclusion principle that no more than one fermion can occupy the same particle state.This is a restriction which cannot be described without considering permutation symmetry.
(f) Holes Dirac showed [1] that there is symmetry between the occupied and unoccupied states for fermions, based on where ε ≡ ∂ ∂ v p is the velocity vector.We note that the field equation is nonlinear in the presence of a pair potential.This equation can be used to derive, and obtain the time dependent version of the Ginzburg-Landau (GL) equation [3].This topic will be treated separately.
For a many-fermion system, fermion field operators ( ) This equation is also nonlinear in the presence of a pair potential ( ) ik φ .
indices.(e) The Second Quantization for FermionsMany fermions can be treated by using the complex dynamical operators ( )