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Chapter#3: Fundamental of Quantum Mechanics

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}<em>i\middle|\hat{A}\middle|\mathrm{\Psi}_i\right\rangle\left\langle\mathrm{\Psi}_j\middle|\mathrm{\Psi}_j\right\rangle=}\sum</em>{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

    \[\Delta L_z \Delta \phi \geq \hbar\]

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