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
$$\left|\left\langle\emptyset\middle|\mathrm{\Psi}\right\rangle\right|^2\le\left\langle\emptyset\middle|\emptyset\right\rangle\left\langle\mathrm{\Psi}\middle|\mathrm{\Psi}\right\rangle$$
(ii) Triangle inequality: For ant two state functions
$$\sqrt{\left\langle\emptyset+\mathrm{\Psi}\middle|\mathrm{\Psi}+\emptyset\right\rangle}\ \le\ \sqrt{\left\langle\emptyset\middle|\emptyset\right\rangle}+\sqrt{\left\langle\mathrm{\Psi}\middle|\mathrm{\Psi}\right\rangle}$$
(iii) Orthogonal states: Any two state functions are said to be orthogonal if
$$\left\langle\mathrm{\Psi}\middle|\emptyset\right\rangle=0$$
(iv) Orthonormal states: The state functions are said to be orthonormal if these are orthogonal as well as normalized to unity.
$$\left\langle\emptyset\middle|\mathrm{\Psi}\right\rangle=0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left\langle\emptyset\middle|\emptyset\right\rangle=1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left\langle\mathrm{\Psi}\middle|\mathrm{\Psi}\right\rangle=1\ \$$
Q.3.3 Define Hermitian and skew-Hermitian Operators.
Answer: An operator is said to be Hermitian if,
$${\hat{A}}^\dag=\hat{A} Or equivalently \left\langle\emptyset\middle|\hat{A}\middle|\mathrm{\Psi}\right\rangle={\left\langle\mathrm{\Psi}\middle|\hat{A}\middle|\emptyset\right\rangle}^\ast$$
An operator is said to be skew-Hermitian if,
$${\hat{A}}^\dag=-\hat{A} Or equivalently \left\langle\emptyset\middle|\hat{A}\middle|\mathrm{\Psi}\right\rangle=-\left\langle\mathrm{\Psi}\middle|\hat{A}\middle|\emptyset\right\rangle$$
Q.3.4 Deduce Heisenberg Uncertainty Principle from Generalized Uncertainty Relation.
Answer: From Generalized Uncertainty relation,
$$\Delta A\Delta B\geq \frac{1}{2}\left | <[\hat{A},\hat{B}]> \right |$$
For position-momentum relation $\left[\hat{x},\ {\hat{p}}_x\right]=i\hbar$ we have
$$\Delta x \Delta P_x\geq \frac{\hbar}{2}$$
Which is Heisenberg Uncertainty Principle.
Q.3.5 what conditions must parameter ‘a’ and operator satisfy so that the operator is unitary?
Answer: For ‘a’ to be real and $\hat{A}$ to be Hermitian, $\hat{U}$ is unitary i.e.
$${\hat{U}}^\dag=\left(e^{ia\hat{A}}\right)^\dag=e^{-ia\hat{A}}=U-1$$
Q.3.6 Define matrix representation of Kets.
Answer: The ket vector can be expanded in terms of base kets as;
The ket vector can be represented by a column vector as follows:
Q.3.7 Define expectation value of an observable valid for discrete and continuous spectra.
Answer: The ket vector can be expanded in terms of base kets as;
$$\left|\mathrm{\Psi}\right.\left.\ \right\rangle=\sum_{i=1}^{\infty}{a_i\left|\emptyset_i\right.\left.\ \right\rangle}$$
The ket vector can be represented by a column vector as follows:
$$\left|\mathrm{\Psi}\right.\left.\ \right\rangle=\left(\begin{matrix}\left\langle\emptyset_1\middle|\mathrm{\Psi}\right\rangle\\left\langle\emptyset_2\middle|\mathrm{\Psi}\right\rangle\\left\langle\emptyset_i\middle|\mathrm{\Psi}\right\rangle\\end{matrix}\right)$$
Q.3.8 what is analogy between quantum mechanical commutation bracket and classical Poisson Bracket?
Answer: Analogy between quantum mechanical commutation bracket and classical Poison Bracket is given below:
$$\left{A,\ B\right}_{classical\ Poison\ Bracket\ }\rightarrow1iћA,B$$
Q.3.9 Define following:
(i) Orthogonal matrix (ii) Unitary matrix (iii) Trace of a matrix
Answer: (i) Orthogonal Matrix: A square matrix A us said to be orthogonal if its transpose is equal to its inverse:
$$A^T=A^{-1}$$
OR equivalently,
$$AA^T=A^TA=I$$
(ii) Unitary Matrix: A matrix A corresponding to operator is said to be unitary if its inverse is equal to its Hermitian conjugate:
$${\hat{A}}^\dag={\hat{A}}^{-1}$$
Or equivalently,
$${\hat{A}}^\dag\hat{A}=\hat{A}{\hat{A}}^\dag=I$$
(iii) Trace of a matrix: Trace of a matrix is defined as sum of diagonal elements of matrix. It is also called spur of a matrix.
For example,
$$A=\left[\begin{matrix}a_{11}&a_{12}\a_{21}&a_{22}\\end{matrix}\right]$$
Then $Trace\ A=\ a_{11}+a_{22}$
Q.3.10 show that trace of an operator does not depend on basis in which it is expressed.
Answer: By definition,
$$Tr\left(\hat{A}\right)=\sum_{i}\left\langle\mathrm{\Psi}_j\middle|\hat{A}\middle|\mathrm{\Psi}_j\right\rangle$$
$$\Rightarrow Tr\left(\hat{A}\right)=\sum_{i}^{} \left\langle\mathrm{\Psi}_j\right. \left | \right |\mathrm{\Psi}_i \left\langle\mathrm{\Psi}_i\middle|\hat{A}\middle|\mathrm{\Psi}_i\right\rangle\left\langle\mathrm{\Psi}_i\right.\left.\ \right|\left|\ \left.\mathrm{\Psi}_j\right\rangle\right.$$
$$\Rightarrow Tr\left(\hat{A}\right)=\sum_{i}{\left\langle\mathrm{\Psi}_i\middle|\hat{A}\middle|\mathrm{\Psi}_i\right\rangle\left\langle\mathrm{\Psi}_j\right.\left.\ \right|\left|\mathrm{\Psi}_i\right.\left.\ \right\rangle\left\langle\mathrm{\Psi}_i\right.\left.\ \right|\left|\mathrm{\Psi}_j\right.\left.\ \right\rangle}$$
$\Rightarrow Tr\left(\hat{A}\right)=\sum_{i}{\left\langle\mathrm{\Psi}i\middle|\hat{A}\middle|\mathrm{\Psi}_i\right\rangle\left\langle\mathrm{\Psi}_j\middle|\mathrm{\Psi}_j\right\rangle=}\sum{i}\left\langle\mathrm{\Psi}_i\middle|\hat{A}\middle|\mathrm{\Psi}_i\right\rangle$ Which completes the proof.
Q.3.11 Define norm of vector and normalized vector.
called norm of ‘u’ and denoted by $\left | u \right |$ . So,
$$\left | u \right |= \sqrt{\left(u,\ u\right)}$$
Normalized vector: If norm of a vector is unity, then it is called normalized vector. Any vector ‘v’ which is not normalized can be normalized by using formula:
$$U=\frac{v}{\left | v \right |}$$
Q.3.12 Define transformation matrix. What is unity transformation?
Answer: We can transform from the representation with application of matrix ‘U’ we can get reverse with matrix . Here U is called transformation matrix U-1. The matrix transforming one orthonormal set into another is unitary and the transformation of this type is known as unitary transformation.
Q.3.15 Show that $\delta\left(x\right)=\lim_{\eta\rightarrow\infty} \frac{sin\ \eta x}{\pi\ x}$
Answer: By definition of delta function,
$$\delta\left(x\right)=\frac{1}{2\pi}\int_{-\infty}^{\infty}{e^{ikx}\ dk}$$
$$\Rightarrow\delta\left(x\right)=\frac{1}{2\pi} \lim_{\eta\rightarrow\infty} \int_{-\eta}^{\eta}{e^{ikx}\ dk=}\frac{1}{2\pi} \lim_{\eta\rightarrow\infty} \frac{2\sin{\eta\ x}}{x}$$
$$\Rightarrow\delta\left(x\right)= \lim_{\eta\rightarrow\infty} \frac{\sin{\eta\ x}}{\pi\ x}$$
Q.3 16 Why are Hermitian operators associated with observable in quantum mechanics?
Answer: Hermitian operators associated with observable in quantum mechanics due to following reasons:
- Eigen values of Hermitian operators are real.
- Eigen functions of Hermitian operators are orthonormal.
- Product of commuting Hermitian operators is Hermitian.
Q.3.17 Name three pairs of canonically conjugate quantities.
Answer: A rectangular coordinate of a particle and corresponding component of momentum are said to be canonically conjugate to each other. There are three pairs of canonically conjugate variables:
- Position of particle and corresponding momentum.
$$\Delta x \Delta p\geq \hbar$$
- Energy of particle and time at which it is measured
$$\Delta E \Delta t \geq \hbar$$
- A component of angular momentum and its angular position in perpendicular plane
$$\Delta L_z \Delta \phi \geq \hbar$$
