Quantum Mechanics-I

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Chapter#2: Mathematical Tools of Quantum Mechanics

Team Quanta gladly presents all possible short questions of BS Physics book Quantum Mechanics-I‘s Chapter#2: Mathematical Tools of Quantum Mechanics.

Q.2.1 what are mathematical requirements that a wave function must satisfy to represent a physical system?

Answer: Wave functions that are physically acceptable and their derivatives must be finite, continuous and single valued represent a physical system.

Q.2.2 what is physical significance of wave function?

Answer: he wave function Ψ(x, t) has no physical significance directly. Its modulus square  has physical meaning, which gives probability density i.e. the quantity \left|\mathrm{\Psi}\left(x,\ t\right)\right|^2\ dx  probability of finding the particle at time ‘t’ in a distance dx. The total probability of finding the system somewhere in space is always unity. Note that the wave functions \mathrm{\Psi}\left(x,\ t\right) and e^{ia}\mathrm{\Psi}\left(x,\ t\right) represent same state, a being a real constant.

Q.2.3 some wave functions are given below:

    \[\Psi_1\left(x\right)=3sin\pi x,\ \ \ \ \Psi_2\left(x\right)=x^2,\ \ \ \ \ \ \Psi_3\left(x\right)=4-\left|x\right|\]

Which of following function are physically acceptable?

Answer: Only the function \mathrm{\Psi}_1\left(x\right)=3sin\pi x is physically acceptable since this function and its derivatives are finite, continuous, single valued and integer able.

\mathrm{\Psi}_2\left(x\right)=x^2 is neither finite nor square integer able.

\mathrm{\Psi}_3\left(x\right)=4-\left|x\right| is not continuous, or finite or square integer able.

Q.2.4 what is superposition principle? Give an example from macroscopic world.

Answer: The state of a system cannot be represents by a single wave function; it can be represented by superposition of many wave functions. According to this principle, \mathrm{\Psi}_1\left(\overline{r},\ t\right)\ \ \mathrm{\Psi}_2\ (\overline{r},\ t) if are solutions of Schrodinger wave equation, then their linear combination a\mathrm{\Psi}_1\left(\overline{r},t\right)+b\mathrm{\Psi}_2(\overline{r},\ t) is also solution of Schrodinger wave equation, where a & b are complex numbers.

Example: The state of vibrating string can be represented by a single wave or superposition of many waves.

Q.2.5 how do we mathematical represent dynamical variables of classical mechanics in quantum mechanics?

Answer: According to one of the postulates of quantum mechanics, every dynamical variable of classical mechanics is represented by Hermitian operator whose Eigen values are real. Some operators of quantum mechanics are listed below:

X-component of linear momentum in operator form is {\hat{p}}_x=-i\hbar \frac{\partial}{\partial x}

Total energy in operator form is \hat{E}=i\hbar \frac{\partial}{\partial t}

Q.2.6 when Hamiltonian has discrete and continuous spectra?

Answer: In case of bound states, Hamiltonian has discrete spectrum of values and for unbound states continuous spectrum.

Q.2.7 Define compatible and non-compatible operators.

Answer: If there exists a complete set of simultaneous Eigen functions of two linear operators A & B, then the operators are said to be compatible. The operators which are not compatible are called non-compatible or incompatible operators.

Q.2.8 If A and B are compatible A and  has degenerate Eigen values. Whether simultaneous Eigen functions of A & B are complete?

Answer: Among degenerate states of A , only a subset of them are Eigen functions of B Hence set of Eigen functions which are simultaneous Eigen functions of A & B  is not complete.

Q.2.9 Stationary states exist only for time independent potentials. Explain.

Answer: Shrodinger wave equation for time independent potential is,

    \[-\frac{\hbar}{2m} \nabla^2\mathrm{\Psi}\left(\overline{r},\ t\right)+V\left(\overline{r}\right)\mathrm{\Psi}\left(\overline{r},t\right)= i\hbar \frac{\partial\Psi(\ \overline{r},\ t)}{\partial t}\]

Solutions to this equation are,

    \[\mathrm{\Psi}\left(\overline{r},\ t\right)=\emptyset(\ \overline{r}\ )e-iEt\hbar\]

This solution of  wave equation for a time independent potential is called a stationary state. The reason is that probability density is independent of time:

    \[\left|\mathrm{\Psi}(\overline{r},\ t)\right|^2=\ \mathrm{\Psi}^\ast\left(\overline{r},\ t\right)\mathrm{\Psi}\left(\overline{r},t\right)=\emptyset^\ast\left(\ \overline{r}\right)e^{iEt/\hbar} \emptyset\left(\ \overline{r}\ \right)e^{-iEt/\hbar}=\left|\emptyset\left(\overline{r}\right)\right|^2\]

Q.2.10 Can we derive  wave equation from 1st principles?

Answer: We cannot derive this equation from basic principles. Only we can postulate it. Wave packets offer a formal tool to obtain this equation.

Q.2.11 Define probability density and probability current density.

Answer: Probability density is defined as;

    \[\rho=\ \mathrm{\Psi}^\ast\left(\ \overline{r},\ t\right)\mathrm{\Psi}\left(\ \overline{r},\ t\right)=\left|\mathrm{\Psi}\left(\overline{r},\ t\right)\right|^2\]

And probability current density is given by relation;

    \[\vec{J}=\frac{\hbar}{2im_o}(\psi ^<em>\Delta \psi -\psi \Delta \psi^</em>)\]

Both quantities are related by equation of continuity.

    \[\frac{\partial\rho}{\partial t}+\nabla.\vec{J}=0\]

Q.2.12 when an observable is said to be constant of motion?

Answer: If  is independent of time and commutes with Hamiltonian, then observable  is said to be constant of motion. From time development of expectation values,

    \[\frac{d}{dt}<a href="http://ccc">=\frac{1}{i} <[\hat{A}, \hat{H}]>+< \frac{\sigma A}{\sigma t} ></a>\]

If \left[\hat{A},\ \hat{H}\right]=0 & <\frac{\partial\hat{A}}{\partial t}>=0 , then \frac{d}{dt}<\hat{A}> =0\ \Rightarrow <\hat{A}>=constant

Q.2.13 what is essence of correspondence principle?

Answer: Classical limit can be described by limit ћ⇾0. In this limit,

    \[\lim_{\hbar\rightarrow 0}Quantum Mechanics \ \rightarrow \ Classical Mechanics\]

This is called essence of correspondence principle.

Q.2.14 what is wave mechanics?

Answer: Representing the formulism of quantum mechanics in continuous basis yields an Eigen value problem in form of  wave equation;

    \[-\frac{\hbar}{2m}\nabla^2\mathrm{\Psi}+V\mathrm{\Psi}=E\mathrm{\Psi}\]

This formulation of quantum mechanics is called wave mechanics.

Q.2.15 The Eigen values of Hermitian operator are real. What you can say about Eigen values of anti-Hermitian operator?

Answer: The Eigen values of anti-Hermitian operator are either pure imaginary or zero.

Q.2.16 what is importance of operators in quantum mechanics?

Answer: The importance of operators in quantum mechanics lies in fact that each dynamical variable of motion of a system can be represented by a linear operator and the Eigen values of that operator give the result of a precise measurement of dynamical variable represented by that operator.

Q.2.17 what are normalized and orthonormal Eigen functions?

Answer: A function Ψ(x) is said to be normalized if;

    \[\int{\mathrm{\Psi}^\ast\left(x\right)\mathrm{\Psi}\left(x\right)dx=1}\]

Two functions  are said to be orthogonal if;

    \[\int\mathrm{\Psi}_m^\ast\left(x\right)\mathrm{\Psi}_n\left(x\right)dx=0\]

If set of Eigen functions is orthogonal as well as normalized, then we say that it is an orthonormal set of Eigen vectors.

Q.2.18 what is necessary and sufficient condition for two dynamical variables to be measured simultaneously?

Answer: The necessary and sufficient condition for two dynamical variables to be measured simultaneously is that the operators corresponding to these variables must commute.

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