Introduction to Mathematical Physics/Quantum mechanics/Postulates

State space

The first postulate deal with the description of the state of a system. Template:IMP/postulat

The space $\mathscr{H}$ have to be precised for each physical system considered. Template:IMP/exmp Quantum mechanics substitutes thus to the classical notion of position and speed a function $\psi (x)$ of squared summable. A element $\psi (x)$ of $\mathscr{H}$ is noted $|\psi >$ using Dirac notations. Template:IMP/exmp Template:IMP/exmp Template:IMP/label Template:IMP/exmp 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} Template:IMP/postulat Template:IMP/postulat Template:IMP/postulat Template:IMP/postulat Template:IMP/rem Template:IMP/rem

Other representations

Other representations can be obtained by unitary transformations. Template:IMP/defn Template:IMP/prop Template:IMP/pf Template:IMP/prop Template:IMP/pf

Heisenberg representation

We have seen that evolution operator provides state at time $t$ as a function of state at time $0$: Template:IMP/eq Let us write ${\varphi }_S$ the state in Schrödinger representation and ${\varphi }_H$ 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: Template:IMP/eq with Template:IMP/eq 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 $t=0$ : ${\varphi }_H={\varphi }_S(0)$. This allows us to adapt the postulate to Heisenberg representation: Template:IMP/postulat Note that if $A_S$ is the operator associated to a physical quantity $𝒜$ in Schrödinger representation, then the relation between $A_S$ and $A_H$ is: Template:IMP/eq Operator $A_H$ depends on time, even if $A_S$ does not. Template:IMP/postulat Spectral decomposition principle stays unchanged: Template:IMP/postulat The relation with Schrödinger is described by the following equality: Template:IMP/eq As $U$ is unitary: Template:IMP/eq Postulate on the probability to obtain a value to measurement remains unchanged, except that operator now depends on time, and vector doesn't. Template:IMP/postulat This equation is called Heisenberg equation for the observable. Template:IMP/rem

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 $|{\psi }_S>$ a state in Schrödinger representation and $|{\psi }_I>$ a state in interaction representation.\index{interaction representation} Template:IMP/eq with Template:IMP/eq Template:IMP/postulat If $A_S$ is the operator associated to a physical quantity $𝒜$ in Schrödinger representation, then relation between $A_S$ and $A_I$ is: Template:IMP/eq So, $A_I$ depends on time, even if $A_S$ does not. Possible results postulate remains unchanged. Template:IMP/postulat 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: Template:IMP/postulat 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.