Introduction to Mathematical Physics/Quantum mechanics/Postulates

secespetat

State spaceEdit

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

Postulate: (Description of the state of a system) To each physical system corresponds a complex Hilbert space   with enumerable basis.

The space   have to be precised for each physical system considered.

Example: For a system with one particle with spin zero in a non relativistic framework, the adopted state space   is  . It is the space of complex functions of squared summable (relatively to Lebesgue measure) equipped by scalar product:

 

This space is called space of orbital states.\index{state space}

Quantum mechanics substitutes thus to the classical notion of position and speed a function   of squared summable. A element   of   is noted   using Dirac notations.

Example:

For a system constituted by a particle with non zero spin \index{spin} , in a non relativistic framework, state space is the tensorial product   where  . Particles with entire spin are called bosons;\index{bosons} Particles with semi-entire spin are called fermions.\index{fermions}

Example: For a system constituted by   distinct particles, state space is the tensorial product of Hilbert spaces   ( ) where   is the state space associated to particle  .

exmppauli

Example: For a system constituted by   identical particles, the state space is a subspace of   where   is the state space associated to particle  . Let   be a function of this subspace. It can be written:

 

where  . Let   be the operator permutation \index{permutation} from   into   defined by:

 

where   is a permutation of  . A vector is called symmetrical if it can be written:

 

A vector is called anti-symetrical if it can be written:

 

where   is the signature \index{signature} (or parity) of the permutation  ,   being the number of transpositions whose permutation   is product. Coefficients   and   allow to normalize wave functions. Sum is extended to all permutations   of  . Depending on the particle, symmetrical or anti-symetrical vectors should be chosen as state vectors. More precisely:

  • For bosons, state space is the subspace of   made by symmetrical vectors.
  • For fermions, state space is the subspace of   made by anti-symetrical vectors.

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

Schrödinger representationEdit

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

Postulate:

(Description of physical quantities) Each measurable physical quantity   can be described by an operator   acting in  . This operator is an observable.

Postulate: (Possible results) The result of a measurement of a physical quantity   can be only one of the eigenvalues of the associated observable  .

Postulate: (Spectral decomposition principle) When a physical quantity   is measured in a system which is in state normed  , the average value of measurement is   :

 

where   represents the scalar product in  . In particular, if  , probability to obtain the value   when doing a measurement is:

 

Postulate: (Evolution) The evolution of a state vector   obeys the Schrödinger\footnote{The Autria physicist Schrödinger first proposed this equation in 1926 as he was working in Zurich. He received the Nobel prize in 1933 with Paul Dirac for their work in atomic physics.} equation:

 

where   is the observable associated to the system's energy.

Remark: State a time   can be expressed as a function of state a time  :

 

Operator   is called evolution operator.\index{evolution operator} It can be shown that   is unitary.\index{unitary operator}

Remark: When operator   doesn't depend on time, evolution equation can be easily integrated and it yields:

 

with

 

When   depends on time, solution of evolution equation

 

is not :

 

secautresrep

Other representationsEdit

Other representations can be obtained by unitary transformations.

Definition: By definition ([#References

Property: If   is hermitic, then operator   is unitary.

Proof: Indeed:

 
 
 

Property: Unitary transformations conserve the scalar product.

Proof: Indeed, if

 

then

 

Heisenberg representationEdit

We have seen that evolution operator provides state at time   as a function of state at time  :

 

Let us write   the state in Schrödinger representation and   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   and equal to the corresponding state in Schrödinger representation for   :  . This allows us to adapt the postulate to Heisenberg representation:

Postulate: (Description of physical quantities) To each physical quantity and corresponding state space   can be associated a function   with self adjoint operators   in   values.

Note that if   is the operator associated to a physical quantity   in Schrödinger representation, then the relation between   and   is:

 

Operator   depends on time, even if   does not.

Postulate: (Possible results) Value of a physical quantity at time   can only be one of the points of the spectrum of the associated self adjoint operator  .

Spectral decomposition principle stays unchanged:

Postulate: (Spectral decomposition principle) When measuring some physical quantity   on a system in a normed state  , the average value of measurements is   :

 

The relation with Schrödinger is described by the following equality:

 

As   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.

Postulate: (Evolution) Evolution equation is (in the case of an isolated system):

 

This equation is called Heisenberg equation for the observable.

Remark: If system is conservative (  doesn't depend on time), then we have seen that

 

if we associate to a physical quantity at time   operator   identical to operator associated to this quantity in Schrödinger representation, operator   is written:

 

Interaction representationEdit

Assume that hamiltonian   can be shared into two parts   and  . In particle,   is often considered as a perturbation of   and represents interaction between unperturbed states (eigenvectors of  ). Let us note   a state in Schrödinger representation and   a state in interaction representation.\index{interaction representation}

 

with

 

Postulate: (Description of physical quantities) To each physical quantity in a state space   is associated a function   with self adjoint operators   in   values.

If   is the operator associated to a physical quantity   in Schrödinger representation, then relation between   and   is:

 

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

Postulate:

When measuring a physical quantity   for a system in state  , the average value of   is:

 

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:

Postulate: (Evolution) Evolution of a vector   is given by:

 

with

 

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.