Eigen function and eigen value,normalization and orthogonal, quantum mechanical operator and postulates of quantum mechanics

Eigen value and eigen value

\(\psi\) = A sin\(\frac{2\pi x}{\lambda}\)

\(\frac{d\psi}{dx}\) = A \(\frac{2\pi}{\lambda}\) cos\(\frac{2\pi x}{\lambda}\)

= \(\frac{2\pi A}{\lambda}\) cos \(\frac{2\pi x}{\lambda}\)

\(\frac{d}{dx}\) is not eigen operator

\(\frac{d^2\psi}{dx^2}\) = \(\frac{d}{dx}\) [\(\frac{d}{dx}\) A sin\(\frac{2\pi x}{\lambda}\)]

=\(\frac{d}{dx}\) [\(\frac{2\pi A}{\lambda}\)cos \(\frac{2\pi x}{\lambda}\)]

=\(\frac{2\pi A}{\lambda}\) [\(\frac{d}{dx}\)cos \(\frac{2\pi x}{\lambda}\)]

=\(\frac{2\pi A}{\lambda}\) (-)\(\frac{2\pi}{\lambda}\)sin\(\frac{2\pi x}{\lambda}\)

= - \(\frac{4\pi^2}{\lambda^2}\) Asin\(\frac{2\pi x}{\lambda}\)

= -\(\frac{4\pi^2}{\lambda^2}\) \(\psi\)

Here, \(\psi\) is called eigen function or A sin\(\frac{2\pi x}{\lambda}\) is called eigen function and-\(\frac{4\pi^2}{\lambda^2}\) is called eigen value\(\frac{d^2\psi}{dx^2}\) is eigen operator,\(\frac{d\psi}{dx}\) is not eigen operator. A function which when operated by an operator gives a certain multiple of the function is called eigen function.

The multiple value obtained is called eigen value or any value which is obtained when a function is operated by an operator to give the eigen function is called value or eigen value is the coefficient of a function when it is operated by eigen operator.

Operator: It is the matheamtical instruction. Operator is anything it allows to do so.\(\frac{d}{dx}\) is a differential operator\(\frac{d}{dx}\)(x²) = 2x \(\int\) integraion operator \(\int\)x².dx = \(\frac{x^3}{3}\) + C

The operator which when operate to a function and give eigen value is called eigen operator.

The solution of Schrodinger wave equation should have eigen value to be significant. Such type of wave functions are finite, single valued continuous and normalised. This type of wave function is also called well-behaved wave function.

The concept of normalization and orthogonal

If a particle show dual nature then its property can be explained by wave function. The amplitude of the wave is denoted by \(\psi\) and it is called amplitude function. The square of amplitude function is the measure of probability of finding the particle whch is called probability amplitude. The value of \(\psi\) may be real or imaginary. So, the value of \(\psi\)²may also be real or imaginary. However, the probability of finding a particle at a point within a space is certain. The probability of certainty is unity. Hence, we use \(\psi\)\(\psi\)^*instead of \(\psi\)²where \(psi\)^*is complex conjugate of \(\psi\). Complex conjugate of wave funciton i s written by the sign of 'i'.

i.e if \(\psi\) = a + ib then\(\psi\)^*= a- ib

The probaility of finding a particle at a point within a space is given by \(\int\) \(\psi\)\(\psi\)^*dx.dy.dz

If a wave obey the relation then the wave is said to be normalised and the operation is called normalization.

\(\int\) \(\psi\)\(\psi\)^*dx.dy.dz = 1

\(\int_0^a\)\(\psi\)\(\psi\)^*dx.dy.dz = 1

\(\int_-\propto^\propto\)\(\psi\)\(\psi\)^*dx.dy.dz = 1

\(\int_all space\)\(\psi\)\(\psi\)^*dx.dy.dz = 1

\(\oint\)\(\psi\)\(\psi\)^*dx.dy.dz = 1

Also,dx.dy.dz = d\(\tau\)

In all cases, the wave function may not be found to be normalised. In such conditions, the wave function is multiplied by a constant which is called normalization constant. Here, it is to be noted that like a wave function, the product of wave function with constant is also a solution of a particle or function. It's conjugate function is\(\psi\)^*. If\(\oint\)\(\psi\)\(\psi\)^* d\(\tau\) is not normalized \(\psi\) is multiplied by a constant A. Then the complex conjugate of wave function is A\(\psi\)^*. Then probability function becomes normalized.

i.e\(\oint\)A \(\psi\)A \(\psi\)^*dx.dy.dz = 1

or,A²\(\oint\)\(\psi\)\(\psi\)*dx.dy.dz = 1

or,\(\oint\)\(\psi\)\(\psi\)^*dx.dy.dz = \(\frac{1}{A^2}\)

or, A = \(\frac{1}{\sqrt(\oint\)\psi\)\psi\)^*dx.dy.dz)}\)

Here, A is normalization constant

In general, if \(\psi\)_iand \(\psi\)_jare two acceptable wave function then for normalization, following criteria should be fulfilled.

\(\oint\)\(\psi\)_i\(\psi\)_j^*dx.dy.dz = 1

\(\oint\)\(\psi\)_j\(\psi\)_i^*dx.dy.dz = 1

If these wave function fulfill the following criteria then these wave functions are said to be mutually orthogonal.

\(\oint\)\(\psi\)_i\(\psi\)_j^*dx.dy.dz = 0

\(\oint\)\(\psi\)_j\(\psi\)_i^*dx.dy.dz = 0

Quantum mechanical operator

An operator is a mathematical instruction or symbol which is carried out to do something on a function to obtain another function. Mathematically, Operator (one function) = another function

Example, \(\frac{d}{dx}\) is differential operator

\(\int\) is integration operator

\(\log\) is \(\log\) operator

\(\hat{H}\) or H is Hamiltonian operator

\(\triangledown\)²is Laplacian operator

The operator operate on a function that function is called operand and the coming function is callled result of operation.

Example \(\frac{d}{dx}x²= 2x

In the above example \(\frac{d}{dx}\) is operator, x²is operand and 2x is result of operation.

Linear operator

An operator is said to be linear operator if it obeys the following relation.

\(\hat{A}\) (f+g) =\(\hat{A}\)f +\(\hat{A}\)g Here, f and g are any two function.

\(\hat{A}\) (c₁+ c₂) = c₁\(\hat{A}\)f + c₂\(\hat{A}\)g Here c₁and c₂are constant

Example: Taking square is linear operator or not?

Square of (2+3) = (2+3)²= 5²= 25

= 2²+ 3² = 4 +9 = 13

25≠ 13 So , taking square is not linear operator.

Laplacian operator (\(\triangledown\)²)

Mathematically laplacian operator is defined as\(\triangledown\)²= \(\frac{\delta^2}{\delta x^2}\)+\(\frac{\delta^2}{\delta y^2}\) +\(\frac{\delta^2}{\delta z^2}\)

In terms of Laplacian operator the Schrodinger wave equation is expressed as

\(\frac{\delta^2\psi}{\delta x^2}\) +\(\frac{\delta^2\psi}{\delta y^2}\) +\(\frac{\delta^2\psi}{\delta z^2}\) + \(\frac{8\pi^2m(E-V)\psi}{h^2}\) = 0

or,\(\triangledown\)² \(\psi\) + \(\frac{8\pi^2m(E-V)\psi}{h^2}\) = 0

Here,\(\triangledown\)²is Laplacian operator. It is two degree differential operator.

Hamiltonian operator (\(\hat{H}\))

Hamiltonian operator is the energy operator for a system. It includes kinetic energy as well as potential energy. It is denoted by \(\hat{H}\) operator.

\(\hat{H}\) = (v - \(\frac{h^2}{8\pi^2 m}\)\(\triangledown\)²)

In terms of Hamiltonian operator, the Schrodinger wave equation can be expresses as,

\(\hat{H}\) \(\psi\) = E\(\psi\)

Hermitian operator

Let U₁and U₂are two suitable functions of x. Let A is operator. The operator is said to be Hermitian operator if the following relation holds good.

\(\int\) U₁^*A U₂ d\(\tau\) = \(\int\) U₂^* A U₁d\(\tau\)

Here the sign Asterik indicates the complex conjugate of another corresponding function.

or

The properties of Hermitian operators are as follows:

1. They have always red Eigen value.
2. Their eigen functions are normalized and mutually orthogonal.
3. Their eigen functions form a complete set.

Postulates of quantum mechanics

Postulates are a kind of assumptios on the basis of which a theory is developed and on the basis of the developed theory, the validity of assumptions are tested. If these are fonund to be true then these are regarded as postulates. The postulates of quantum mechanices are as follows:

1. The state of a quantum mechanical system is completely described by a wave function.

Here, the terim complete information means as much information as possible with the consistent of uncertaninty proinciple. For instance a wave function \(\psi\) (x,t) denotes the co-ordinates of various particle with time. The wave function give the complete information about behaviour of the particle. The wave function \(\psi\) (x) dentoes the stationary wave function regardless of time. The square fo wave funciton interpretes the probability of finding the particle in the given volume.

The probability of finding a particle within the whole space is unity.

\(\oint\) \(\psi\)^*(x) \(\psi\) (x) d\(\tau\) =1

To express the probalbilty of finding the particle in 3-D space, trip;e integrate sign is written.

\(\oint\)\(\oint\)\(\oint\) \(\psi\)^*(x,y,z) \(\psi\)(x,y,z) dxdydz = 1

2. Each observable in classical mechanics correpond to a linear operator in quantum mechanics.

Measurable dynamical variable such as position momentum, angular momentum, energy etc are called observable. Classical description of a system to quantum mechanical description is linked by operator.

3. The time development of a system can be described by the evolution of state vector in time given by the time development Schrodinger wave equaton.

\(\psi\)_(t)= \(\psi\)_(o).e^{-\(\frac{i\hat{H}t}{\hslash}\)}

.e^{-\(\frac{i\hat{H}t}{\hslash}\)} is time development operator

4. In general, the measurement of physical observable disturb the system immediately after the measurement of the system in a state which is an eigen state of corresponding operator and the measured value is one of the eigen values of this operator.

|\(\psi\)⟩ = C₁ |\(\psi\)⟩ + C₂ |\(\psi\)⟩ + .......+ C_n|\(\psi\)⟩

\(\hat{A}\)|\(\psi\)⟩ = C₁a_1|\(\psi\)⟩+ C₂a₂|\(\psi\)⟩ + .......+ C_na_n|\(\psi\)⟩

5. State vectors of a system are represented by normlised vectors belonging to the linear vector space of state vectors. All normalised vectors belonging to this space represent physicallly realizable states.

If the set of state vectors|\(\psi\)₁⟩,|\(\psi\)₂⟩, ......|\(\psi\)_n⟩ represent a basis then all normalised vectors|\(\psi\)⟩ are of the form,|\(\psi\)⟩ = E_iC_i|\(\psi\)_i⟩

6. The eigen state of operator corresponding to any physical observable constitute a complete set. For every observable 'q' there is an operator\(\hat{Q}\) and a complete set of eigen states|\(\psi\)⟩ such that

\(\hat{Q}\) |\(\psi\)_i⟩ = q_i|\(\psi\)_i⟩

7, The average of large number of measurement is callled expectation value of the obsevalbe.

Example, The expectation value of kinetic energy is denoted by⟨T⟩

8. The commutator of operator in quantum mechanics are related to corresponding to classical poisson bracket.

[u,v]⇒ i\(\hslash\) {u,v} p