Quantum Mechanics Quiz

1. Which of the following wave functions can be solutions of Schrodinger's equation for all values of x?

$\psi=A \sec x$

$\psi=A \tan x$

$\psi=A e^{x^2}$

$\psi=Ae^{-x^2}$

2. Consider the wave function $\psi(x,t)=A e^{-\lambda x^2-i\omega t}$, where A, $\lambda$ and $\omega$ are positive real constants. The value of A for which the wave function is normalized

$(\frac{\lambda}{\pi})^{1/4}$

$(\frac{2\lambda}{\pi})^{1/4}$

$(\frac{\lambda}{\pi})^{1/2}$

$(\frac{2\lambda}{\pi})^{1/4}$

3. Consider the wave function $\psi(x,t)=Ae^{-\lambda |x|} e^{-i\omega t}$, where A, $\lambda$ and $\omega$ are positive real constants. The value of A for which the wave function is normalized

$\sqrt{\lambda}$

$\frac{1}{\sqrt{\lambda}}$

$\sqrt{2\lambda}$

$\frac{2}{\sqrt{\lambda}}$

4. A particle of mass m is in the state $\psi(x,t)=A e^{-a(mx^2/\hbar)} e^{-iat}$, where A and a are positive real constants. For what potential energy function $V(x)$ does $\psi(x,t)$ satisfy Schrodinger equation?

$ma^2x^2$

$2ma^2x^2$

$3ma^2x^2$

$4ma^2x^2$

5. A particle of mass m is in the state $\psi(x,t)=(\frac{2am}{\pi\hbar})^{1/4} e^{-a(mx^2/\hbar)} e^{-iat}$. What are the values of standard deviation in $x$ ($\sigma_x$) and standard deviation in $p$ ($\sigma_p$)?

$\sigma_x=\sqrt{\frac{\hbar}{4am}}$ and $\sigma_p=\sqrt{am\hbar}$

$\sigma_x=\sqrt{\frac{\hbar}{am}}$ and $\sigma_p=\sqrt{am\hbar}$

$\sigma_x=\sqrt{\frac{\hbar}{4am}}$ and $\sigma_p=\sqrt{4am\hbar}$

$\sigma_x=\sqrt{\frac{\hbar}{2am}}$ and $\sigma_p=\sqrt{2am\hbar}$

6. The expectation values of position and momentum are

$\frac{d}{dt}\langle x \rangle =-\frac{\langle p \rangle}{m}$ and $\frac{d}{dt}\langle p \rangle =\langle \frac{\partial V}{\partial x}\rangle$

$\frac{d}{dt}\langle x \rangle =\frac{\langle p \rangle}{m}$ and $\frac{d}{dt}\langle p \rangle =\langle \frac{\partial V}{\partial x}\rangle$

$\frac{d}{dt}\langle x \rangle =\frac{\langle p \rangle}{m}$ and $\frac{d}{dt}\langle p \rangle =\langle -\frac{\partial V}{\partial x}\rangle$

$\frac{d}{dt}\langle x \rangle =-\frac{\langle p \rangle}{m}$ and $\frac{d}{dt}\langle p \rangle =\langle -\frac{\partial V}{\partial x}\rangle$

7. The dimension of wave function for one dimensional system is

$L^{-1/2}$

$L^{-3/2}$

$L^{-1}$

$L$

8. The SI unit of wave function for a three dimensional system is

$m^{-1/2} s^{-2}$

$m^{-3/2} s{^-1}$

$m^{-3/2}$

$m^{-1/2}$

9. Consider a one-dimensional particle which is confined within the region $0\leq x \leq a$ and whose wave function is $\psi(x,t)=\sin(\pi x/a)e^{-i\omega t}$. The probability of finding the particle in the interval $ a/4 \leq x \leq 3a/4 $ is

$\frac{1+\pi}{\pi}$

$\frac{2+\pi}{\pi}$

$\frac{2+\pi}{2\pi}$

$\frac{\pi}{2}$

10. The probability current density for a particle of mass m with wave function, $\psi(x)=A e ^{ikx} + B e^{-ikx}$ is

$\frac{\hbar k}{m}(|A|^2+|B|^2)$

$\frac{\hbar}{m}(|A|^2+|B|^2)$

$\frac{\hbar k}{m}(|A|^2-|B|^2)$

$\frac{\hbar}{m}(|A|^2-|B|^2)$

11. The operator $(\frac{d}{dx}+x)(\frac{d}{dx}-x)$ is equivalent to

$\frac{d^2}{dx^2}-x^2-1$

$\frac{d^2}{dx^2}-x^2+1$

$\frac{d^2}{dx^2}-x^2$

$\frac{d^2}{dx^2}-x-1$

12. The eigenfunction of the momentum operator $\hat{p}$ with eigenvalue $p_0$ is

$\frac{1}{\sqrt{2\pi\hbar}}e^{-ip_0x/\hbar}$

$\frac{1}{\sqrt{2\pi\hbar}}e^{ip_0x/\hbar}$

$\frac{1}{\sqrt{2\pi\hbar}}e^{ip_0/\hbar}$

$\frac{1}{\sqrt{2\pi\hbar}}e^{-p_0x/\hbar}$

13. The eigenfunction of the position operator $\hat{x}$ with eigenvalue $x_0$ is

$\delta(x-x_0)$

$\delta(x)$

$e^{-ix x_0/\hbar}$

$e^{ix x_0/\hbar}$

14. Which of the following operators are Hermitian?

Position

Momentum

Total energy

All of these

15. The eigenvalues of a Hermitian operator are

real

purely imaginary

zero

complex numbers

16. The wave function of a particle is given by $\psi(x)=\frac{3}{5}\phi_1(x) + \frac{4}{5}\phi_2(x)$, where $\phi_1(x)$ and $\phi_2(x)$ are eigenfunctions with corresponding energy eigen values $-1eV$ and $-2eV$ respectively. The energy of the particle in the state $\psi$ is

$-\frac{41}{25}eV$

$-\frac{11}{5}eV$

$-\frac{36}{25}eV$

$-\frac{7}{5}eV$

17. The commutator of position operator $\hat{x}$ and momentum operator $\hat{p}$ is

$[\hat{x},\hat{p}]=i\hbar$

$[\hat{x},\hat{p}]=-i\hbar$

$[\hat{x},\hat{p}]=\hbar$

$[\hat{x},\hat{p}]=-\hbar$

18. The commutator $[\hat{x}^n,\hat{p}]$ is

$i\hbar nx^{n+1}$

$-i\hbar nx^{n+1}$

$-i\hbar nx^{n-1}$

$i\hbar nx^{n-1}$

19. The Hermitian conjugate of the operator $\frac{d}{dx}$ is

$-\frac{d}{dx}$

$\frac{d}{dx}$

$-i\frac{d}{dx}$

$i\frac{d}{dx}$

20. If A linear operator $\hat{O}$ acts on two orthonormal states of a system $\psi_1$ and $\psi_2$ as per following: $\hat{O}\psi_1=\psi_2$ and $\hat{O}\psi_2=\frac{1}{\sqrt{2}}(\psi_1+\psi_2)$. The system is in a mixture of two states defined by $\psi=\frac{1}{\sqrt{2}}\psi_1+ \frac{i}{\sqrt{2}}\psi_2$. The expectation values of $\hat{O}$ in the state $\psi$ is

$\frac{1}{2\sqrt{2}}(1+i(\sqrt{2}+1))$

$\frac{1}{2\sqrt{2}}(1-i(\sqrt{2}+1))$

$\frac{1}{2\sqrt{2}}(1+i(\sqrt{2}-1))$

$\frac{1}{2\sqrt{2}}(1-i(\sqrt{2}-1))$

21. Given two Hermitian operators $\hat{x}$ and $\hat{p}$, which of the following combination would be Hermitian?

$\hat{x}\hat{p}$

$\hat{x}+i\hat{p}$

$\hat{x}\hat{p}+\hat{p}\hat{x}$

$\hat{x}\hat{p}-\hat{p}\hat{x}$

22. The uncertainty relation between two operators $\hat{A}$ and $\hat{B}$ is

$\Delta A\Delta B \leq \frac{1}{2}|\langle [\hat{A},\hat{B}]\rangle|$

$\Delta A\Delta B \geq |\langle [\hat{A},\hat{B}]\rangle|$

$\Delta A\Delta B \leq |\langle [\hat{A},\hat{B}]\rangle|$

$\Delta A\Delta B \geq \frac{1}{2}|\langle [\hat{A},\hat{B}]\rangle|$

23. The uncertainty relation between position and momentum operator is

$\Delta x\,\Delta p \leq \frac{\hbar}{2}$

$\Delta x\,\Delta p \geq \frac{\hbar}{2}$

$\Delta x\,\Delta p \geq \frac{1}{2}$

$\Delta x\,\Delta \leq \frac{1}{2}$

24. The position-momentum uncertainty for a Gaussian wave packet is

$\Delta x\,\Delta p = 0$

$\Delta x\,\Delta p = \hbar$

$\Delta x\,\Delta p = \frac{\hbar}{2}$

$\Delta x\,\Delta p \geq \frac{\hbar}{2}$

25. A particle of mass $m$ is in a one dimensional potential, $V(x)=\begin{cases}
0, & 0\leq x \leq L\\
\infty, & \text{otherwise}
\end{cases}$. The normalized energy eigen functions and energy eigen values are respectively

$\psi_n(x)=\sqrt{\frac{1}{L}}\sin(\frac{n\pi}{L}x)$ and $E_n=\frac{n^2\pi^2\hbar^2}{mL^2}$

$\psi_n(x)=\sqrt{\frac{2}{L}}\sin(\frac{n\pi}{L}x)$ and $E_n=\frac{n^2\pi^2\hbar^2}{2mL^2}$

$\psi_n(x)=\sqrt{\frac{1}{L}}\sin(\frac{n\pi}{L}x)$ and $E_n=\frac{n^2\pi^2\hbar^2}{8mL^2}$

$\psi_n(x)=\sqrt{\frac{4}{L}}\sin(\frac{n\pi}{L}x)$ and $E_n=\frac{n^2\pi^2\hbar^2}{mL^2}$

26. A particle of mass $m$ is in a one dimensional infinite square well of width $L$. The expectation values $\langle x \rangle$ and $\langle p \rangle$ for the ground state is

$\frac{L}{4}$ and $0$

$\frac{L}{3}$ and $1$

$\frac{L}{2}$ and $0$

$L$ and $1$

27. Consider a particle moving freely between $x=0$ to $x=a$ inside an infinite square well potential. The uncertainties product $\Delta x \Delta p$ in the ground state is

$\frac{\hbar}{2}\sqrt{\frac{\pi^2}{3}-2}$

$\frac{\hbar}{2}\sqrt{\frac{\pi^2}{3}+2}$

$\frac{\hbar}{2}\sqrt{\frac{\pi^2}{3}-1}$

$\frac{\hbar}{2}\sqrt{\frac{\pi^2}{3}+1}$

28. A particle in the infinite square well of width a has its initial wave function an even mixture of first two stationary states, $\psi(x,0)=A[\psi_1(x)+\psi_2(x)]$. The normalized wave function $\psi(x,t)$ is

$\frac{1}{\sqrt{a}}[\sin(\frac{\pi x}{a}) e^{-\pi^2\hbar^2/2ma^2} + \sin(\frac{2\pi x}{a}) e^{-4\pi^2\hbar^2/2ma^2}]$

$\frac{2}{\sqrt{a}}[\sin(\frac{\pi x}{a}) e^{-\pi^2\hbar^2/2ma^2} + \sin(\frac{2\pi x}{a}) e^{-4\pi^2\hbar^2/2ma^2}]$

$\frac{1}{\sqrt{a}}[\sin(\frac{\pi x}{a}) e^{-\pi^2\hbar^2/2ma^2} + \sin(\frac{2\pi x}{a}) e^{-2\pi^2\hbar^2/2ma^2}]$

$\frac{2}{\sqrt{a}}[\sin(\frac{\pi x}{a}) e^{-\pi^2\hbar^2/2ma^2} + \sin(\frac{2\pi x}{a}) e^{-2\pi^2\hbar^2/2ma^2}]$

29. A particle in the infinite square well has its wave function an even mixture of first two stationary states, $\psi(x)=\sqrt{\frac{1}{3}}\psi_1(x)+\sqrt{\frac{2}{3}}\psi_2(x)$. Identify the correct statement.

$\langle x \rangle =\frac{L}{2}$ and $\langle E \rangle=\frac{3 \pi^2 \hbar^2}{2mL^2}$

$\langle x \rangle =\frac{2L}{3}$ and $\langle E \rangle=\frac{ \pi^2 \hbar^2}{2mL^2}$

$\langle x \rangle =\frac{L}{4}$ and $\langle E \rangle=\frac{3 \pi^2 \hbar^2}{8mL^2}$

$\langle x \rangle =\frac{L}{3}$ and $\langle E \rangle=\frac{8 \pi^2 \hbar^2}{2mL^2}$

30. The normalized ground state wave function of a one dimensional harmonic oscillator is

$(\frac{m\omega}{\pi\hbar})^{1/2} e^{-\frac{m\omega}{2\hbar}x^2}$

$(\frac{m\omega}{\pi\hbar})^{1/4} e^{-\frac{m\omega}{2\hbar}x^2}$

$(\frac{m\omega}{\pi\hbar})^{1/2} e^{-\frac{m\omega}{\hbar}x^2}$

$(\frac{m\omega}{\pi\hbar})^{1/4} e^{-\frac{m\omega}{\hbar}x^2}$

31. The expectation values $\langle x \rangle$ and $\langle p \rangle$ for the ground state of the harmonic oscillator are respectively

0 and 1

1 and 0

1 and 1

0 and 0

32. The uncertainties product $\Delta x \Delta p$ for the ground state of harmonic oscillator is

$\Delta x \Delta p=\hbar$

$\Delta x \Delta p=\frac{\hbar}{2}$

$\Delta x \Delta p=\frac{\hbar}{4}$

$\Delta x \Delta p=\frac{3\hbar}{2}$

33. The expectation values of kinetic energy and potential energy for ground state of the harmonic oscillator are respectively

$\langle T \rangle=\frac{\hbar}{4}$ and $\langle V \rangle=\frac{\hbar}{4}$

$\langle T \rangle=\frac{\hbar}{2}$ and $\langle V \rangle=\frac{\hbar}{4}$

$\langle T \rangle=\frac{\hbar}{4}$ and $\langle V \rangle=\frac{\hbar}{2}$

$\langle T \rangle=\frac{\hbar}{2}$ and $\langle V \rangle=\frac{\hbar}{2}$

34. The zero point energy of the harmonic oscillator is

$0$

$\hbar$

$\frac{\hbar}{2}$

$\frac{\hbar}{4}$

35. A particle is in a state which is a superposition of the ground state $\phi_0$ and the first excited state $\phi_1$ of a one dimensional harmonic oscillator. The state is given by $\phi=\frac{1}{\sqrt{5}}\phi_0 +\frac{2}{\sqrt{5}}\phi_1$. The expectation value of energy of the particle is

$\frac{11}{10}\hbar\omega$

$\frac{13}{10}\hbar\omega$

$\frac{13}{5}\hbar\omega$

$\frac{11}{5}\hbar\omega$

36. A particle with energy $E$ is incident on a potential given by $V(x)=\begin{cases}
0, & x< 0\\
V_0, & x \geq 0
\end{cases}$. The wave function of the particle for $V<E_0$ in the region $x>0$ (in terms of positive constants A,B and k) is

$Ae^{kx}+Be^{-kx}$

$Ae^{-kx}$

$Ae^{ikx}+Be^{-ikx}$

zero

37. A particle is moving inside a infinite cubical well $V(x)=\begin{cases}
0, & \text{if}\,\, 0\leq x\leq a, 0\leq y\leq a, 0\leq z\leq a\\
\infty, & \text{otherwise}
\end{cases}$. The ground state wave function and ground state energy are

$\psi_1=(\frac{2}{a})^{3/2} \sin\frac{\pi x}{a} \sin\frac{\pi y}{a} \sin\frac{\pi z}{a}$, $E_1=\frac{3\pi^2\hbar^2}{2m a^2}$

$\psi_1=(\frac{2}{a})^{1/2} \sin\frac{\pi x}{a} \sin\frac{\pi y}{a} \sin\frac{\pi z}{a}$, $E_1=\frac{\pi^2\hbar^2}{2m a^2}$

$\psi_1=(\frac{3}{a})^{3/2} \sin\frac{\pi x}{a} \sin\frac{\pi y}{a} \sin\frac{\pi z}{a}$, $E_1=\frac{2\pi^2\hbar^2}{2m a^2}$

$\psi_1=(\frac{2}{a})^{1/2} \sin\frac{\pi x}{a} \sin\frac{\pi y}{a} \sin\frac{\pi z}{a}$, $E_1=\frac{\pi^2\hbar^2}{m a^2}$

38. For a quantum particle confined in a cubic box of side $L$, the ground state energy is given by $E_1$. The energy of the first excited state is

$E_1$

$2E_1$

$4E_1$

$6E_1$

39. Three identical non-interacting particles, each of spin $\frac{1}{2}$ and mass $m$, are moving in a one dimensional infinite potential well of width $a$. The energy of the lowest energy state of the system is

$\frac{\pi^2\hbar^2}{ma^2}$

$\frac{2\pi^2\hbar^2}{ma^2}$

$\frac{3\pi^2\hbar^2}{ma^2}$

$\frac{4\pi^2\hbar^2}{ma^2}$

40. A system of 8 non-interacting electrons is confined by a three dimensional potential $V(r)=\frac{1}{2}m\omega^2r^2$. The ground state energy of the system is

$6\hbar\omega$

$12\hbar\omega$

$18\hbar\omega$

$24\hbar\omega$