Describing the same thing in two different ways

Clearly, the two have something in common, but to really appreciate why they are equivalent, it is best to adopt Dirac's formalism, which entails them both. Here, both the wave function and state vector are produced from the state ket |Ψ> in the following way:

Ψ(x) = <x|Ψ>
Ψ_j = <j|Ψ>

QΨ(x) = <x|Q_S|Ψ> = (integral d³x') <x|Q_S|x'><x'|Ψ> = (integral d³x') Q(x,x')Ψ(x')
[ QΨ ]_i = <i|Q_H|Ψ> = (sum over j) <i|Q_H|j> <j|Ψ> = (sum over j) Q_ijΨ_j

Schrödinger picture

HΨ(x,t) = i(h-bar) (partial d by dt)Ψ(x,t)

H|Ψ,t> = i(h-bar)d|Ψ,t>

exp(A) = 1 + A + (1/2!)A² + (1/3!)A³ + ...

|Ψ,t> = U(t)|Ψ,0>

<φ,t|Ψ,t> = <φ,0|U(t)⁺U(t)|Ψ,t> = <φ,0|Ψ,0>

<φ,t|Q_S|Ψt> = <φ,0|U(t)⁺Q_SU(t)|Ψ0>

Heisenberg picture

<φ,0|Q_h(t)|Ψ,0> = <φ|Q_h(t)|Ψ>

Given this, one can produce a time evolution equation for operators Q_h(t) in Heisenberg's picture. Consider the time derivative of the above expression

i(h-bar)<φ|dQ_h(t)|Ψ>
= i(h-bar)d<φ|Q_h(t)|Ψ>
= i(h-bar)d<φ,t|Q_S|Ψ,t> = i(h-bar)<φ,t|Q_SH|Ψ,t> -i(h-bar)<φ,t|hQ_S|Ψ,t>
= <φ,t|[Q_S,H ]|Ψ,t>
= <φ|U(t)⁺[ Q_S(t),H ]U(t)|Ψ>
= <φ|[ Q_H(t),H ]|Ψ>

i(h-bar)dQ_H(t) = [ Q_H(t),H ]

Why bother?

There are several approaches to Quantum Mechanics, called pictures. All of them come up with the same results; they only differ in their internal organization. The most commonly taught elementary form is the Schrodinger Picture, devised by Erwin Schrodinger. It is so dominant that the others are not typically taught below the graduate level. But what's the difference?

First, let's consider the various things that go into an expression in quantum mechanics.

• You have a state of interest. This is represented by a vector.
• You have an observable quantity of interest, for example, the momentum. This is represented by a linear operator acting on the vector.
• You have time dependence.

The pictures differ in where the time dependence goes.

Looked at from a straightforward point of view, it is obviously the state of the particle which varies in time. The way of getting the momentum out of the state does not vary in time -- the definition of momentum hasn't changed in time, certainly! This is the Schrodinger Picture, and it is fine for many purposes.

But that's not the only possibility. In classical mechanics, most commonly you express the state simply as the set of functions of the observable characteristics as a function of time. For example, you can give the center of mass position in each dimension as a function of time, x(t), y(t), z(t). Other observables such as the momentum or angular momentum can be given their own functions, and there are relationships between them.

From this point of view, it is most natural to consider the observable characteristics as what vary in time. So you put the time dependence in the linear operators and use the state as an initial condition. This is the Heisenberg Picture.

But that's not all! In Quantum Mechanics, there is a lot of trivial time dependence. If the potential is constant in time (i.e. the system is not pushed on from outside) then the way to find out the behavior of the system is to identify all of the states with definite energy (called energy eigenstates*). If you use these states as the basis for your state, then letting time advance is very simple: just let the complex phase of each component advance at the rate of E/ℏ. This is very simple to execute once you have the solution.

Now suppose the potential is not constant in time, but rather has one component which stays the same and one component which varies. No longer is the time dependence just that simple spinning**. However, the simple spinning does comprise a large fraction of the time dependence, if the time varying part is small. The Interaction Picture separates the trivial time dependence from the nontrivial time dependence, by hiding the trivial time dependence in the observables, and putting the more complicated time dependence in the state.

Note that the Interaction Picture itself is not an approximation. It is technically applicable in every situation. However, it is most useful when the perturbation is small. In that case, the Interaction Picture simplifies the task of making the appropriate approximation. See Path Integral.

There are other more esoteric pictures of quantum mechanics. Those of you who know of them, feel free to add them!

One interesting side-note, though, is that when one attempts to incorporate relativity, thus yielding Quantum Field Theory, the only practical picture is that of Heisenberg, because only it treats time just like it treats space.

unperson notes: since evolution in quantum mechanics is unitary, you can just think of it as a rotation (or the generalization there of, anyway). The difference between the Schroedinger and Heisenberg pictures is nothing but the difference between an active transformation and a passive transformation.

* see eigenvector for some clarification on why they would be called this. Keep in mind that the Energy is found by applying the Hamiltonian, a linear operator (a.k.a. matrix), to the state vector.

** This is not the same thing as spin, as in fermions or bosons.