Understanding quantum mechanics "picture" terms

Question

I was reading various sources and a have some questions.

1. The "Schrödinger picture" is the same thing as "Schrödinger wave formulation"?
2. Is "Heisenberg picture" the same thing as "Heisenberg matrix formulation of quantum mechanics"?

From Wikipedia:

Heisenberg Picture: It stands in contrast to the Schrödinger picture in which the operators are constant, instead, and the states evolve in time. The two pictures only differ by a basis change with respect to time-dependency, which corresponds to the difference between active and passive transformations.

Matrix Mechanics: It is equivalent to the Schrödinger wave formulation of quantum mechanics, as manifest in Dirac's bra–ket notation.

If "Heisenberg picture" and "matrix mechanics" is the same (my second question), then how it is both equivalent and in contrast to the same thing?

Answer 1

I do not know for question 1 but for question 2:

No, Heisenberg picture is an interpretation in which the time dependance of a system comes from the operators instead of the wavefunctions (like in the Schrodinger picture).

In other words in the Schrodinger picture we have: $$\left|\psi\right\rangle=\left|\psi(t)\right\rangle$$ and the Shrodinger equation reads as usual: $$-i\hbar\frac{\partial}{\partial t}\left|\psi(t)\right\rangle=H\left|\psi(t)\right\rangle$$ And the expectation value is given by:

$$\left\langle O(t)\right\rangle=\left\langle\psi(t)|O|\psi(t)\right\rangle$$

Whereas in the Heisenberg picture, the wavefunctions do not depend on time and the time evolution is dictated by the operators which carry the time dependance. Thus, the time dependance of the expectation value of an operator is given by:

$$\left\langle O(t)\right\rangle=\left\langle\psi|O(t)|\psi\right\rangle$$

And since the time evolution is dictated by the Hamiltonian, the operators follow the equation:

$$\frac{d}{dt}O(t) = \frac{i}{\hbar}[O(t), H] + \frac{\partial}{\partial t}O(t)$$

The matrix formulation of quantum mechanics can be used both in the Schrodinger picture than in the Heisenberg picture.

Answer 2

1. The "Schrödinger picture" is the same thing as "Schrödinger wave formulation"?

Yes

2. Is "Heisenberg picture" the same thing as "Heisenberg matrix formulation of quantum mechanics"?

Yes.

Matrix Mechanics: It is equivalent to the Schrödinger wave formulation of quantum mechanics, as manifest in Dirac's bra–ket notation.

Matrix mechanics (i.e. the Heisenberg picture) is equivalent to the Schrödinger formulation, in that they both have the same physics content, but they're still different formulations.

They are different ways of saying the same thing, so they're equivalent in that they're saying the same thing, and they're in contrast to each other in that they're different ways to say it.

Answer 3

When we say that two things are "equivalent" in physics, we generally mean the following:

$A$ and $B$ are equivalent if one can be transformed into the other using operations that don't affect physical meaning.

For example, we know that a measurement of the distance between two points is equivalent to a measurement of the time it takes light to travel between two points. This is because light always travels at the speed $c$, so therefore for any time interval $t$, there is a corresponding distance $d=ct$. In this case, multiplying or dividing by the speed of light is an operation that doesn't change the physical meaning of the measurement in the framework of relativity (this is, in fact, part of the rationale for setting $c=1$ in relativity, to emphasize the equivalence of distance and time intervals).

You'll notice, in the above, that two things can be equivalent without being literally the same thing. Distance is not the same thing as time (for proof, see the impossibility of traveling backwards in time), but every distance interval can be converted into a time interval. Any two non-identical things can be contrasted, even if they are equivalent. For example, we could say, "distance intervals, which are always nonzero for spacelike-separated events, stand in contrast to time intervals, which can be zero for spacelike-separated events." So there's no contradiction in saying that two things are equivalent and can be contrasted.

In our particular case, the Schrodinger and Heisenberg pictures are equivalent because any statement made in the Schrodinger picture can be transformed into a unique statement in the Heisenberg picture, by executing what is effectively a change of basis (which doesn't change the physical meaning). They can be contrasted because they are not literally the same thing - in the Schrodinger picture, the time-dependence of the statement is carried by the wavefunctions, while in the Heisenberg picture, the time-dependence of the statement is carried by the operators.