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|ϕn〉 = an |ϕn〉
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 an.
Postulate 4 (a): When the physical quantity A is measured on a system in the state |ψ〉, the probability P(an) of obtaining the non-degenerate eigen value an of the corresponding observable
Postulate 4 (b): When the physical quantity A is measured on a system in the state |ψ〉.
The probability P(an) of obtaining the eigen value an of the corresponding observable A is,
Where gn is the degree of degeneracy of an and(i = 1, 2, 3, &, gn) is orthonormal set of vector which forms a basis in the eigen subspace and associated with eigenvalue an 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
(d) Square integrable.
The expectation value of operator A is given
For continuous variable-
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,
where a(p) being expansion coefficient of |p〉.
If any function Ψ(x) can be expressed as a linear combination of state function ϕn
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 〈ϕi | ϕj〉, δij , then
(a) check whether is normalized or not
(b) write down normalized wavefunction.
(c) it is given H |ϕn〉 = (n + 1)ℏω| ϕn〉 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.
(c) It is given that
H|ϕn〉 = (n + 1) ℏω, where n = 0,1, 2, 3, 4, ....
H |ϕ1〉 2ℏω and H |ϕ2〉 = 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 : |ϕ1〉 |ϕ2〉
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 ϕ0 and ϕ1 are the normalised eigenfunctions with energy E0 and E1 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 E0 ,E1
(c) Probability of getting
Example 16: (a) Plot ΨI (x) = A1e-|x| ; -∞ < x < ∞
(c) Discuss why ψI is not the solution of Schrödinger wave function rather ψII is solution of Schrödinger wave function.
(a) ψI (x) = A1e+x ; x < 0
ψII (x) = A1e-x ; x > 0
The plot is given by
(c) Both the function ψI and ψII are single valued, continuous, square integrable but ψI is not differentiable at x = 0 , rather ψII is differentiable at x = 0
So, ψII can be solution of Schrödinger wave function but ψI 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 |ϕn〉 = n2∈0|ϕn〉
(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 = (〈H2〉 - 〈H〉2)1/2
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, 2a0 , 2a0 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, 2a0, respectively?
(b) What is the expectation value of A ?
λ2 = λ3 = 2a0 i.e., λ = 2a0 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.