Team Quanta presents all possible short questions of BS Physics book Quantum Mechanics-I’s Chapter#3: Fundamental of Quantum Mechanics.
Q.3.1 what are dimensions of Hilbert space?
Answer: Dimensions of Hilbert space of square integrable functions is infinite, since each wave function can be expanded in terms of infinite number of linearly independent functions.
Q.3.2 Define following:
(i)Schwarz inequality (ii) Triangle inequality (iii) Orthogonal states (iv) Orthonormal states
Answer: (i) Schwarz inequality: For ant two state functions
(ii) Triangle inequality: For ant two state functions
(iii) Orthogonal states: Any two state functions are said to be orthogonal if
(iv) Orthonormal states: The state functions are said to be orthonormal if these are orthogonal as well as normalized to unity.
Q.3.3 Define Hermitian and skew-Hermitian Operators.
Answer: An operator is said to be Hermitian if,
An operator is said to be skew-Hermitian if,
Q.3.4 Deduce Heisenberg Uncertainty Principle from Generalized Uncertainty Relation.
Answer: From Generalized Uncertainty relation,
For position-momentum relation we have
Which is Heisenberg Uncertainty Principle.
Q.3.5 what conditions must parameter ‘a’ and operator satisfy so that the operatoris unitary?
Answer: For ‘a’ to be real and to be Hermitian, is unitary i.e.
Q.3.6 Define matrix representation of Kets.
Answer: The ket vector can be expanded in terms of base kets as;
The ket vector can be represented by a column vector as follows:
Q.3.7 Define expectation value of an observable valid for discrete and continuous spectra.
Answer: The ket vector can be expanded in terms of base kets as;
The ket vector can be represented by a column vector as follows:
Q.3.8 what is analogy between quantum mechanical commutation bracket and classical Poisson Bracket?
Answer: Analogy between quantum mechanical commutation bracket and classical Poison Bracket is given below:
Q.3.9 Define following:
(i) Orthogonal matrix (ii) Unitary matrix (iii) Trace of a matrix
Answer: (i) Orthogonal Matrix: A square matrix A us said to be orthogonal if its transpose is equal to its inverse:
OR equivalently,
(ii) Unitary Matrix: A matrix A corresponding to operator is said to be unitary if its inverse is equal to its Hermitian conjugate:
Or equivalently,
(iii) Trace of a matrix: Trace of a matrix is defined as sum of diagonal elements of matrix. It is also called spur of a matrix.
For example,
Then
Q.3.10 show that trace of an operator does not depend on basis in which it is expressed.
Answer: By definition,
Which completes the proof.
Q.3.11 Define norm of vector and normalized vector.
called norm of ‘u’ and denoted by . So,
Normalized vector: If norm of a vector is unity, then it is called normalized vector. Any vector ‘v’ which is not normalized can be normalized by using formula:
Q.3.12 Define transformation matrix. What is unity transformation?
Answer: We can transform from the representation with application of matrix ‘U’ we can get reverse with matrix . Here U is called transformation matrix U-1. The matrix transforming one orthonormal set into another is unitary and the transformation of this type is known as unitary transformation.
Q.3.15 Show that
Answer: By definition of delta function,
Q.3 16 Why are Hermitian operators associated with observable in quantum mechanics?
Answer: Hermitian operators associated with observable in quantum mechanics due to following reasons:
Eigen values of Hermitian operators are real.
Eigen functions of Hermitian operators are orthonormal.
Product of commuting Hermitian operators is Hermitian.
Q.3.17 Name three pairs of canonically conjugate quantities.
Answer: A rectangular coordinate of a particle and corresponding component of momentum are said to be canonically conjugate to each other. There are three pairs of canonically conjugate variables:
Position of particle and corresponding momentum.
Energy of particle and time at which it is measured
A component of angular momentum and its angular position in perpendicular plane