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**Introduction **

- The quantum mechanical postulates enable us to understand.
- How a quantum state is described mathematically at a given time t.
- How to calculate the various physical quantities from this quantum state.
- Knowing the system’s state at a time t , how to find the state at any later time t .

i.e., how to describe the time evolution of a system.

There are following set of postulates:**Postulate 1:** The state of any physical system is specified, at each time t , by a state vector |ψ(t)〉 in the Hilbert space. |ψ(t)〉 contains all the needed information about the system. Any superposition of state vectors is also a state vector.**Postulate 2: **To every measurable quantity A to be called as an observable or dynamical variable, there corresponds a linear Hermitian operator Aˆ whose eigen vectors form a complete basis A|ϕ_{n}〉 = a_{n} |ϕ_{n}〉**Postulate 3:** The measurement of an observable A may be represented formally by an action of Â on a state vector |ψ(t)〉.

The state of the system immediately after the measurement is the normalized projection on to the eigen subspace associated with a_{n}.**Postulate 4 (a): **When the physical quantity A is measured on a system in the state |ψ〉, the probability P(a_{n}) of obtaining the non-degenerate eigen value a_{n} of the corresponding observable

**Postulate 4 (b): **When the physical quantity A is measured on a system in the state |ψ〉.

The probability P(a_{n}) of obtaining the eigen value a_{n} of the corresponding observable A is,

Where g_{n} is the degree of degeneracy of a_{n} and(i = 1, 2, 3, &, g_{n}) is orthonormal set of vector which forms a basis in the eigen subspace and associated with eigenvalue a_{n} of A.**Postulate 5:** The time evolution of the state vector |ψ(t)〉 is governed by schrodinger equation given by:

where H is Hamiltonion of the system.

The solution of schodinger equation must be**(a)** Single valued and the value must be finite**(b)** Continuous**(c)** Differentiable**(d)** Square integrable.

**Expectation Value**

The expectation value of operator A is given

For continuous variable-

- Error in measurement of A is

**Fourier transformation**

Change in basis from one representation to another representation |p〉 is defined as,

The expansion of Ψ(x) in terms of |p〉 can be written as,

where a (p) can be found as,

In 3D:

where a(p) being expansion coefficient of |p〉.

If any function Ψ(x) can be expressed as a linear combination of state function ϕ_{n}

which is popularly derived from fourier trick.

**Parity operator: **The parity operator defined by its action on the basis.

If ψ(-r) = ψ(r), then state has even parity and

If ψ(-r) = -ψ(r) , then state has odd parity.

So, parity operator have +1 and -1 eigen value.

Representation of postulate (4) in continuous basis.

**Example 14: A state function is given by It is given that 〈ϕ _{i} | ϕ_{j}〉, δ_{ij} , then**

(a) check whether is normalized or not

(b) write down normalized wavefunction.

(c) it is given H |ϕ_{n}〉 = (n + 1)ℏω| ϕ_{n}〉 where n = 0,1, 2, 3, 4, .... .If H is measured on |ψ〉, then what will be measurement and with what probability?

(d) Find the expectation value of H i.e., 〈H〉

(e) Find the error in the measurement of H.

(a) To check normalization, one should verify-

The value of 〈ψ|ψ〉 = 3/2 , so |ψ〉 is not normalized.(b) Now we need to find normalized |ψ〉 let A be normalization constant.

So,(c) It is given that

H|ϕ_{n}〉 = (n + 1) ℏω, where n = 0,1, 2, 3, 4, ....

H |ϕ_{1}〉 2ℏω and H |ϕ_{2}〉 = 3ℏω

when H will be measured on |ψ〉, it will measured either 2ℏω or 3ℏω

The probability of measurement 2ℏω is P(2ℏω) is given by

So, when H is measured on state |ψ〉, the following outcome will come:

Measurement of H on state : |ϕ_{1}〉 |ϕ_{2}〉

Measurement : 2ℏω 3ℏω

Probability : 2/3 1/3(e) The error in measurement in H is given as

**Example 15: The wave function of a particle is given by where ϕ _{0} and ϕ_{1} are the normalised eigenfunctions with energy E_{0} and E_{1} corresponding to ground state and first excited state.**

(a) Find the value of B such that Ψ is normalised.

(b) What are the measurements

(c) What is the probability of getting energy E1

(d) What is 〈E〉

For normalized |ψ〉,(b)Measurement are E_{0},E_{1}(c)Probability of getting

**Example 16: (a) Plot Ψ _{I} (x) = A_{1}e^{-|x|} ; -∞ < x < ∞**

(b)

(c) Discuss why ψ_{I} is not the solution of Schrödinger wave function rather ψ_{II} is solution of Schrödinger wave function.

(a) ψ

_{I}(x) = A_{1}e^{+x}; x < 0

ψ_{II}(x) = A_{1}e^{-x}; x > 0

The plot is given by

(b)(c) Both the function ψ

_{I}and ψ_{II}are single valued, continuous, square integrable but ψ_{I}is not differentiable at x = 0 , rather ψ_{II}is differentiable at x = 0

So, ψ_{II}can be solution of Schrödinger wave function but ψ_{I}is not the solution of Schrödinger wave function.

**Example 17: At time t = 0 , the state vector |ψ(0)〉 is given as, **

It is given that, Hamiltonian is defined as H |ϕ_{n}〉 = n^{2}∈_{0}|ϕ_{n}〉

(a) What is wave function |ψ(t)〉 at later time t.

(b) Write down expression of evolution of |ψ(x, t)|^{2}

(c) Find Δ H

(d) Find the value of Δ HΔt

(b)Evolution of shape of the wave packet

(c)Δ H = (〈H^{2}〉 - 〈H〉^{2})^{1/2}(d)

**Example 18: Consider a one-dimensional particle which is confined within the region 0 ≤ x ≤ a and whose wave function is Find the potential V(x).**

From the fifth postulate:

**Example 19: Eigenvalues of operator A are 0, 2a _{0} , 2a_{0} and corresponding normalized eigen vector are respectively, then if the system is in state then**

(a) When A is measured on system in state then what is the probability of getting value 0, 2a_{0}, respectively?

(b) What is the expectation value of A ?

λ_{2}= λ_{3}= 2a_{0}i.e., λ = 2a_{0}is doubly degenerate.

**Example 20: A free particle which is initially localized in the range -a < x < a is released at time t = 0.(a) Find A such that ψ (x) is normalized.(b) Find ϕ(x) i.e., wave function in momentum space.(c) Find ψ (x, t) i.e., wave function after time t.**

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