Module 1: Quantum Mechanics  2


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1 Quantum Mechanics  Assignment Question: Module 1 Quantum Mechanics Module 1: Quantum Mechanics (a) What do you mean by wave function? Explain its physical interpretation. Write the normalization condition of the wave function. (b) What is the physical meaning of the expectation value of an observable? Write down the basic postulates of quantum mechanics. 0. (a) Show that the momentum and energy operator can be expressed as ˆ ˆ pˆ x i and Eˆ i. x t (b) What do you mean by Eigen function and eigen value? Examine the following wave functions are eigen function of pˆ x or not (i) x ( i) sin x, ( ii) cos x, ( iii) Ae. Hence find the corresponding eigen value. 03. (a) Compute the following commutator bracket and explain physically (i) [x, ] (ii) [x, px ] (iii) [x, p y ] (iv) [z, p x ] (v) [x, H] (vi) [p, H] x (b) Starting from the relation [x, p x ] = i show that [x, p n x ] = i np n1 x and [p x, x n ] = i nx n1 x 04. A wave function (at t = 0) is given by ( x) ae ikx e known as Gaussian wave packet. Determine (a) the normalization constant, (b) the probability density and (c) the probability current density of the wave packet. [NBU(H)003] 05. The wave function of a particle is given by x ( x) Ce, x where C and are constants. Calculate the probability of finding the particle in the region 0 x. [BU(H) 00]. 06. Find the Normalization constant of the onedimensional wave function given by x ( x) C sin( ), 0 x l l [BU(H) 1998, 001]. 0, outside Find the expectation value of xˆ and pˆ. 07. The Wave function of a particle is given by (x) = 3 x, 0 < x < 1 = 0, elsewhere (i) Calculate the probability of finding the particle in the region 0 x 0.5. (ii) What is average position of the particle? 08. The normalized radial wave function for the 1s state of Hydrogen atom is given by 3/ r / a R 10 (r) = (1/ a 0 0) e, where a 0 is the 1s Bohr radius. (a) Prove that the radial probability density is maximum at r = a 0 (b) Calculate the expectation value of the potential energy of the electron in the 1s state of Hydrogen atom. Prepared by Dr. Rajesh Das, Department of Applied Sciences, HITHaldia Page 1
2 Quantum Mechanics  Assignment Question: Module The normalized wave function of a particle on a straight line is given by 1 x ipx ( x) exp ( ) (a) Where is the particle most likely to be found? (b) What is the expectation value of momentum of the particle? x 10. (a) A particle is represented by the wave function ( x) e sin( x), what is the probability that its position to the right of the point x=1. ax (b) A one dimensional wave function is given by ( x) e find the probability of finding a particle between x=1/a and x=/a. 11. The normalized radial wave function for the 1s state of Hydrogen atom is given by 3/ r / a R 10 (r) = (1/ a 0 0) e, where a 0 is the 1s Bohr radius. Prove that the radial probability density is maximum at r = a 0. Calculate the expectation value of the potential energy of the electron in the 1s state of Hydrogen atom. 1. (a) Establish Schrödinger s timedependent equation in onedimension. Hence finally write the Schrödinger s timedependent equation in threedimension. (b) What are the stationary states? Starting from the timedependent Schrödinger s equation derive the Schrödinger s timeindependent equation satisfied by the wave functions of stationary states in 1D using the method of separation of variable. Extend the discussion in three dimension. [B.U(H) 1993] (c) In what respect does the Schrödinger s equation differ from classical wave equation? Explain the term stationary state of quantum mechanical system. In a stationary state E what is the timedependent part of the wave function? [BU(H) 1995] (d) What solutions of Schrödinger s time independent equation are called stationary? When such solutions are obtained? What is the form of the time dependent part of the wave function for stationary state? [BU(H) 1999] (e) Verify that Schrödinger s wave equation is linear. [CU(H) 1993, 97] (f) The general solution of the time dependent 1D Schrödinger s equation is given by ( x, n1 a e n n ie t / n. Examine whether the probability density is independent of time or not. [BU(H) 004] 13. (a) What do you mean by Probability and Probability Current density in a quantum mechanical i system? Show that the Probability Current density is given by J ( r, [ * * ]. m (b) Write down the orthogonality condition for the wave functions. What is kronecker delta? explain its properties with example. Prepared by Dr. Rajesh Das, Department of Applied Sciences, HITHaldia Page
3 Quantum Mechanics  Assignment Question: Module 1 (c) If 1( x, and ( x, are both the solutions of Schrödinger s wave equation for a given potential V(x,, then show that a1 1( x, a ( x, in which a 1 and a are arbitrary constants is also a solution [BU(H) 006] (d) A system has two eigen states 0 and and are the corresponding normalized wave functions. At an instant the system is in a superposed state 1 a1 1 a and a1. (i) Find the value of a if is normalized. (ii) What is the probability that an energy measurement would yield a value of 3 0 (iii) Find out the expectation value. [WBUT007] (e) Consider a1 1 a, where 1 and are orthonormal energy eigenstates of a 1 system corresponding to the energy E 1 and E at t = 0. If is normalized and a 1 (i) Find the value of a. (ii) Find the expectation value of E. Write down the wave function at subsequent time. [WBUT011] (f) The wave function of a particle at a time t is given by ie nt / ient / ( r, a1 1( r) e a( r) e, where 1 and are two normalized wave functions with energies E 1 and E ( E1 E ) ; a 1 and a are constants. Calculate the probability density. Does represent a stationary state? Justify your answer. [BU(H) 004] (g) The stationary state of twolevel system with energy in ev is given by it / ( r, 0.8u 0 ( r) 0.6u1 ( r) e. What is the probability of finding the system in the upper energy level? What is the outcome of energy measurement of the system? [BU(H) 006] 14. Consider a particle in one dimensional potential box having dimension 0 x a with the boundary condition 0 for x 0 and x a and V=0. (a) Write down the free particle Hamiltonian and time independent Schrodinger equation. (b) Derive the normalized wave function and energy eigen values. (c) Explain graphically the different energy levels with corresponding eigen functions and occupation probability. (d) Calculate the value of lowest energy of an electron in onedimensional force free region of length 4Å. (e) The ground state energy of a particle trapped in 1D box is 40eV. What will be the wavelength of a photon due to electronic transition from nd excited state to 1 st excited state and from 1 st excited state to ground state level. 15. The ground state wave function of a particle confined to a onedimensional box with the x ( x) C sin( ), 0 x l dimension L is given by, l. 0, outside Evaluate the normalization constant. What will be the average momentum? Calculate the uncertainties in position and momentum in the state and hence check the validity of the uncertainty relation. [BU(H) 1998] What will be the energy eigen value in these states? 16. A particle of mass m is confined with in the limit l / < x < l/ and it can move only along x axis. No external force acts on the particle. (a) Write down the Hamiltonian of the particle and time independent Schrodinger equation (b) Derive the normalized wave function and energy eigen values. Prepared by Dr. Rajesh Das, Department of Applied Sciences, HITHaldia Page 3
4 Quantum Mechanics  Assignment Question: Module (a) Set up the time independent Schrodinger equation for the particle in three dimensional box with the boundary condition 0 for x 0 and x a, y 0 and y a and z 0 and z a and potential, V(x, y, z) = 0 for 0 < x < l, 0 < y < l, 0 < z < l = for elsewhere (b) Derive the normalized wave function and energy eigen values using separation of variable method. (c) Define degeneracy. Examine the degree of degeneracy of the energy level with energy 3 ml. Show that the energy be 3fold degenerate. 6h 8mL E of a free particle moving in 3dimensional box will Prepared by Dr. Rajesh Das, Department of Applied Sciences, HITHaldia Page 4
5 Quantum Mechanics  Assignment Question: Module 1 Prepared by Dr. Rajesh Das, Department of Applied Sciences, HITHaldia Page 5
6 Quantum Mechanics  Assignment Question: Module 1 Prepared by Dr. Rajesh Das, Department of Applied Sciences, HITHaldia Page 6
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