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Tutorial Problems in Quantum Mechanics: CHEM 21112 - May 2014 1) a) Calculate the De Broglie wave length in nm for a particle of mass 2.1 x 10-34 kg moving at a velocity of 2.0 x 10 8 m s-1. b) Calculate the momentum of an X- ray photon with a wavelength of 0.17 nm. Compare this momentum with the momentum of an electron accelerating through a potential difference of 100 V. 2) The longest wave length of light that induces emission of electrons from a surface of potassium metal is 564 nm. Calculate the work function for potassium in units of joules and electron volts. 3) a) A beam of electrons moves with a velocity of (1000 ± 0.01) m s-1. Using the uncertainty principle, show how precisely the position of an electron can be located at any given instant. b) If the uncertainty in locating a particle moving in one dimension is equaled to its De Broglie wavelength, calculate the fractional uncertainty in its velocity. 4) Define the wave function as applied to quantum mechanical systems. a) Considering a quantum mechanical system with a normalized wave function ϕ write down the general expression for, (α) the total probability of finding the particle (β) the average value of the physical quantity M. (ɣ) the quantum mechanical operator for the square of the linear momentum (i.e. along a one dimensional space x.) 5) Write down the mathematical expressions for the following. i) Normalization factor N for a wave function Ψ. ii) Orthonormality for the two real wave functions Ψ and φ. 6) a) Using the standard symbols, write down the expression for, i) The Schrodinger time independent equation. ii) The quantum mechanical expression for the Hamiltonian operator for a particle moving in three dimensional space. b) A particle in a 1 – D box moves in the length X = 0 to X = L/3. This movement can be described by a wave function. Write down the mathematical expressions for, i) This wave function, ii) The probability of finding the particle in the above length iii) The average value for the observable physical property M if the wave function has been normalized iv) The quantum mechanical Hamiltonian operator if the above particle moves in a 2 –D space under a potential energy of V. 7) a) Based on its classical mathematical expression, derive an expression for Hamiltonian operator for linear motion using standard symbols. b) A particle moves in a 3-D box under a potential energy of V. Using standard symbols write down an expression for Hamiltonian operator for this system. c) Write the quantum mechanical linear momentum operator related to the movement of the particle in (7) (b) above. i) Write down the Eigen value equation corresponding to the statement ii) Solve the differential equation you wrote for the step (c) (i) above and express the solution as a function of x. 8) Consider a particle of mass, m, moving along in a I -D box of length L described by a normalized wave function Ψ = { 2/L}1/2 Sin( nπ/ L) x ( V = 0, if X is zero or greater than zero and equal x and less than L). i) What is the probability of finding the particle between x = L/3 and x = 2L/3. ii) Write down the expression that you would use to find the average value of position of x. iii) Calculate the probability that a particle in the ground state of the above system lies between x = 0 and x = L/3. iv) Sketch the square of the wave function as a function of the position in the first and second excited states and identify the positions of nodes, if any. v) A particle of mass, m, moves in a two dimensional box . The length of each side is L. Write down the expressions for the first three energy levels using the standard symbols and show them on an appropriate sketch. Illustrate the meaning of the degeneracy using the above diagram. 9) a) Calculate the wavelength of radiation emitted when an electron in gallium arsenide potential well of width 10.0 nm falls from its first excited state to ground state. b) Calculate the wavelength of radiation that will be absorbed when an electron from HOMO of the butadiene is excited to the LUMO of it. Single and double bond distances between C and C are 154 pm and 134 pm respectively. c) Calculate the energy separation between n = 1 and n= 2 of nitrogen molecule confined to a one dimensional length of 1 nm. Calculate the value of “n” that corresponds to the thermal energy of nitrogen molecule at 300 K. (The average thermal energy is 1/2 kT). 10) a) Write down the equation for the time independent Schrodinger equation using the standard symbols. b) Write down the classical expression for total energy E in terms of momentum corresponding to a movement of a particle in three dimensional space. Assume that the potential energy V = 0 c) Derive the quantum mechanical expression for the total energy corresponding to the problem above. 11)Based on the knowledge of the wave function, Ψ given below which corresponds to a particle moving in a in 1 D box Ψ = { 2/L}1/2 Sin( nπ/ L) x , a) Write down the expressions for the wave functions of the first and second excited states of the 1 D box system. 12)A cubic box contains 8 electrons and length of a side of it is 1.0 nm. Calculate the energy required in kJ and eV for promotion of an electron from its HOMO to LUMO. [ Hint : Use the knowledge of wave functions and Schrodinger equation corresponding to 1 – D box problem and extend that to 3- D problem. Ignore the inter electronic repulsion.( i.e. V = 0) ] 13) Using the standard form of the wave function Ψ for a particle in the ground state of a one dimensional box, (α ) Find the average value for the square of the linear momentum of the particle px2 in the one dimensional space. ( β ) the probability of finding the particle in the space between x = 0 and x = L/2 14) A particle of mass , m moves in a 2 D rectangular box with length 2 L and width L. Using the standard symbols, write down the expressions for the energy values of first four energy levels of the particle. 15) The 1 - D quantum mechanical linear momentum operator has an eigen function φ with an Eigen value λ i) Write down an Eigen value equation corresponding to the above statement. ii) Solve the differential equation you wrote for the above and express Ψ as a function of x. 16) Using the standard, derive the expression for the quantum mechanical Hamiltonian operator based on its classical mechanical expression. a) The wave function Ψ is a solution of the time independent Schrodinger equation for a particle in one dimensional space with a length L. Prove that -d2 Ψ/dx2 = (n2π2/L2) Ψ. b) Determine the expression for k given that Ψ= A Sin (kx) is a solution to the differential equation written above. 17)A cubic box of edge length of 10.0 nm contains 14 electrons. a) Applying your knowledge of particles in I D box model and hence that of 3 D model, sketch the energy level diagram for the ground state of the system. Neglect the interelectronic repulsions among electrons. b) What do you understand by the term “‘triply degenerate state”? Illustrate this term using the energy level diagram drawn for this system. 18)A particle of mass 1.0 x 10 -27 g moves in an I D box of length 10 nm. The energy value of the particle at a given state is 4.95 x 10 -21 J. a) Determine the quantum number n for this energy level. b) Calculate the frequency of radiation in Hurtz and wave length of radiation in nm required for an electron to excite from the ground energy level to first excited energy level. 19) a) Write down the Eigen value equation based on the statement, “The 1 D quantum mechanical linear operator has an Eigen function ϕ(x) with an Eigen value m”. b) Solve the differential equation wrote for the above step and express ϕ(x) as a function of 1 D variable x. 20) a) If the masses of the nucleus and the electron are “M” and “m” respectively and the total energy and the potential energy of the H atom are “ET” and “V” respectively write a Eigen value equation for the energy of the H atom taking into consideration the motion of both the nucleus and the electron. b) Rearrange the equation you wrote to the step (a) above into the form of a Schrodinger equation and state the problem faced in solving this equation. c) State the Born Oppenheimer approximation and write down the Schrodinger equation in Cartesian coordinates for the motion of the electron of the H atom according to the Born Oppenheimer approximation. Prof. K. A. S. Pathiratne, Department of Chemistry, University of Kelaniya. May 2014.