## Postulates of Quantum Mechanics

The mathematical formulation of quantum mechanics is based on linear operators as may be seen from the postulate given below, often named as the postulates of **quantum mechanics**.

### (i) Postulate 1

The first postulate states that "there is a state vector (or wave-function) associated with every physical state of the system which contains the entire description".

Thus according to first postulate the information about the properties of a system is contained in the function of all coordinates and time that is called the **wave function**. This information may be definite or it may be in statistical form.

### (ii) Postulate 2

For every physically observable there exist a linear Hermitian operator. The **operators** associated with various dynamical variables like energy, momentum, may be obtained as follows :

`\left(\frac{-\ħ^2}{2m}\nabla^2+V\right)\psi=i\ħ\frac{\partial\psi}{\partial t}`

`\hat H\psi=i\ħ\frac{\partial\psi}{\partial t}`

As Hamiltonian 'H' represents energy, the above equation in the form of eigen value equation may be expressed as

`\hat H\psi=\hat E\psi`

### (iii) Postulate 3

The third postulate states that only possible values which a measurement of the observable can yield are given values P_{λ} of the equation

`\hat P\psi_\lambda=\P_\lambda\psi_\lambda`

### (iv) Postulate 4

The fourth postulate gives the rules for extracting information from the **wave function**. The expectation value of a variable of a system in the state Ψ is given by

`<f> = \frac{\int\psi^\ast\hat f\psi dx}{\int\psi^\ast\psi dx}`

The expectation value represent the arithmetic mean over a large number of a simultaneous experiments in identical states Ψ. In general, wave mechanics admits a fluctuation in these measurements, while in classical mechanics it is assumed that every variable is absolutely determine in principle.

### (v) Postulate 5

The Eigen function of the **operators** corresponding to observable forms a complete set.

### (vi) Postulate 6

The wave function or state function of a system evolve in time according to the time-dependent schrodinger equation

`\hat H\psi(r,t)=i\ħ\frac{\partial\psi}{\partial t}`

## Bra-Ket Notation

In **quantum mechanics**, Bra-Ket notation is a standard notation for describing quantum states. The notation use angle brackets '<' and '>' and a vertical bar '|' symbol to denote the scalar product of vectors or the action of a linear function on a vector in a complex vector space. The scalar product is written as

`<\phi\|\psi>`

The right part is called the **ket** vector typically represented as a column vector and written as | Ψ >. The left part is called the **bra** vector. It is the hermitian conjugate of the ket, typically represented as a row vector and is written < φ |.

A **bra** and **ket** with the same label are hermitian conjugate of each other and is also known as the * Dirac notation*.