### 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 Qij acting on states vectors Ψj (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> in the following way:

Ψ(x) = <x>
Ψj = <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|QS> = (integral d3x') <x|QS|x'><x'|Ψ> = (integral d3x') Q(x,x')Ψ(x')
[ ]i = <i|QH> = (sum over j) <i|QH|j> <j|Ψ> = (sum over j) QijΨj
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|QS|Ψt> = <φ,0|U(t)+QSU(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 Qh(t) = U(t)+QSU(t) to obtain
<φ,0|Qh(t)|Ψ,0> = <φ|Qh(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 Qh(t) in Heisenberg's picture. Consider the time derivative of the above expression

i(h-bar)<φ|dQh(t)|Ψ>
= i(h-bar)d<φ|Qh(t)|Ψ>
= i(h-bar)d<φ,t|QS|Ψ,t> = i(h-bar)<φ,t|QSH|Ψ,t> -i(h-bar)<φ,t|hQS|Ψ,t>
= <φ,t|[QS,H ]|Ψ,t>
= <φ|U(t)+[ QS(t),H ]U(t)|Ψ>
= <φ|[ QH(t),H ]|Ψ>
Nothing that QS is constant in time. So the equation of motion in the Heisenberg picture is
i(h-bar)dQH(t) = [ QH(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.