Introduction to Mathematical Physics/Quantum mechanics/Postulates

State space
The first postulate deal with the description of the state of a system.

The space $${\mathcal H}$$ have to be precised for each physical system considered.

Quantum mechanics substitutes thus to the classical notion of position and speed a function $$\psi(x)$$ of squared summable. A element $$\psi(x)$$ of $$\mathcal H$$ is noted $$|\psi>$$ using Dirac notations.

To present the next quantum mechanics postulates, "representations" ([#References|references]) have to be defined.

Schrödinger representation
Here is the statement of the four next postulate of quantum mechanics in Schrödinger representation.\index{Schrödinger representation}

Other representations
Other representations can be obtained by unitary transformations.

Heisenberg representation
We have seen that evolution operator provides state at time $$t$$ as a function of state at time $$0$$:

Let us write $$\phi_S$$ the state in Schrödinger representation and $$\phi_H$$ the state in Heisenberg representation. \index{Heisenberg representation} Heisenberg\footnote{Wener Heisenberg received the Physics Nobel prize for his work in quatum mechanics} representation is defined from Schrödinger representation by the following unitary transformation:

with

In other words, state in Heisenberg representation is characterized by a wave function independent on $$t$$ and equal to the corresponding state in Schrödinger representation for $$t=0$$ : $$\phi_H=\phi_S(0)$$. This allows us to adapt the postulate to Heisenberg representation:

Note that if $$A_S$$ is the operator associated to a physical quantity $$\mathcal A$$ in Schrödinger representation, then the relation between $$A_S$$ and $$A_H$$ is:

Operator $$A_H$$ depends on time, even if $$A_S$$ does not.

Spectral decomposition principle stays unchanged:

The relation with Schrödinger is described by the following equality:

As $$U$$ is unitary:

Postulate on the probability to obtain a value to measurement remains unchanged, except that operator now depends on time, and vector doesn't.

This equation is called Heisenberg equation for the observable.

Interaction representation
Assume that hamiltonian $$H$$ can be shared into two parts $$H_0$$ and $$H_i$$. In particle, $$H_i$$ is often considered as a perturbation of $$H_0$$ and represents interaction between unperturbed states (eigenvectors of $$H_0$$). Let us note $$|\psi_S\mathrel{>} $$ a state in Schrödinger representation and $$|{\psi}_I\mathrel{>} $$ a state in interaction representation.\index{interaction representation}

with

If $$A_S$$ is the operator associated to a physical quantity $$\mathcal A$$ in Schrödinger representation, then relation between $$A_S$$ and $$A_I$$ is:

So, $$A_I$$ depends on time, even if $$A_S$$ does not. Possible results postulate remains unchanged.

As done for Heisenberg representation, one can show that this result is equivalent to the result obtained in the Schrödinger representation. From Schrödinger equation, evolution equation for interaction representation can be obtained immediately:

Interaction representation makes easy perturbative calculations. It is used in quantum electrodynamics ([#References|references]). In the rest of this book, only Schr\"odinge representation will be used.