Describing the same thing in two different ways

History fails me as to how exactly this all came about, but in the early twentieth century there were two schemes for formulating quantum mechanics; on the one hand, there was Erwin Schrödinger's wave mechanics and other the other was Werner Heisenberg's matrix mechanics. The two approches seemed initially very different: the former involved probabilistic picture of the atom, and the latter offered no picture at all - just a load of matrix algebra.
In Schrödinger's idea of quantum mechanics, wave functions Ψ(x) are acted on by linear operators Q(x,x')whose eigenvalues correspond to the observables of the system (and thus doing the physics in this scenario corresponds to solving Sturm-Liouville problems).
Heisenberg's idea was that the eigenvalues are found from matricies Q_ij acting on states vectors Ψ_j (here, the physics is solved by doing linear algebra).

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>

<x| and <j| labeling the respective spaces on which they are defined. Now, the actions of the operators are written

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

Note that I have given the operators different names in each case. The reason why is to follow.
So the operators and states in each case correspond to operators and states in Dirac's formalism; this allows both to be viewed on a common ground. What is interesting is that when it comes to time dependence, the states and operators of the two are different, although in a way that thankfully gives the same physics:

Schrödinger picture

When time dependence is introduced in Schrödinger's wave mechanics, the wave functions Ψ(x) become time dependent wave functions Ψ(x,t). In turn, the corresponding kets in Dirac's notation become time dependent too - |Ψ,t>.
Schrödinger's celebrated wave equation then dictates how the wave functions change with time; this is given by the Hamiltonian operator H acting on the wave functions:

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

Writing out (partial d by dt) every time will become tiring to read, so I'll abbreviate it to d. In Dirac's picture, this equation becomes

H|Ψ,t> = i(h-bar)d|Ψ,t>

Please do not confuse d for an eigenvalue! If you were to define the exponential of an operator A to be

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

(assuming of course that you could in some sense ensure that this infinite series converges. This is a detail most physics choose to ignore, so so shall we)
then you could write the solution of the previous equation as

|Ψ,t> = U(t)|Ψ,0>

for U(t) = exp(-iHt/(h-bar)). Note that U is unitary, by the hermicity of H, and so bra-ket products are preserved under the advancement of time:

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

Also, the some expression of an operator in this basis becomes

<φ,t|Q_S|Ψt> = <φ,0|U(t)⁺Q_SU(t)|Ψ0>

.

Heisenberg picture

Looking at that last equation above, you could consider the states to be constant in time, ie. |Ψ> = |Ψ,0> and the operators Q_h(t) = U(t)⁺Q_SU(t) to obtain

<φ,0|Q_h(t)|Ψ,0> = <φ|Q_h(t)|Ψ>

as an equivalent form of that same equation.
This is exactly what happens when time dependence in Heisenberg's picture is translated into the Dirac formalism. Although the states and operators are different, the inner products and eigenvalues are the same either way, and so the physics is the same as claimed.

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>

Nothing that Q_S is constant in time. So the equation of motion in the Heisenberg picture is

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

Of course, you could start from this and then derive Schrödinger's equation of motion for kets.

Why bother?

Firstly, I suppose, for historical reasons phyics lecturers like to remind us of how ideas progress. By the same token, it's a neat demonstration of how two theories are equivalent.
Another reason would be that it show that there is more than one way to deal with time dependence in the Dirac formalism.

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 Schrodinger picture becomes impractical, because does not treat time and space on the same footing.

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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.

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* 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.