St. Joseph’s College of Arts& Science (Autonomous) St. Joseph’s College Road, Cuddalore – 607001 PCH703T – QUANTUM MECHANICS AND MOLECULAR STRUCTURE

Time : 3 hrs

Max Marks : 75 SECTION - A (20X1=20) Answer ALL Questions

I. Choose the correct answer:1. To what speed must a proton be accelerated for it to have a wavelength of 3.0 cm? (ms-1) b) 2.4 c) 1.3x10-5 d) 2.4x10-1 a) 1.3x10-5 2. Calculate the energy per photon for radiation of wavelength 600 nm (red). a) 210cal b) 210J c) 210 eV d) 210 KJ 3. The fine-structure constant, α, plays a special role in the structure of matter; its approximate value is 1/137. What is the wavelength of an electron travelling at a speed αc, where c is the speed of light? (Note that the circumference of the first Bohr orbit in the hydrogen atom is 331 pm.) a) 3.36m b) 3.36 pm c) 3.36 nm d) 3.36Å 4. The quantum state of a particle moving in a circular path in a plane is given by Ψm(φ) = (1/√2π)elmφ, m = 0, ±1, ±2,..........When a perturbation H1=Pcosφ is applied (P is a constant), what will be the first order correction to the energy of the mth state a) 0 b) P/(2π) c) P/(4π) d) Pm2/(4π2) 5. Determine the commutators of the operators d/dx and 1/x, a) -1

b) −

1 x2

c)

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1 x2

d) x 2

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6. Calculate the de Broglie wavelength of a mass of 1.0 g travelling at 1.0 cm s−1 a) 6.626x1029m b) 6.626x10-30m 28 c) 6.626x10 m d) 6.626x10-29m 7. The uncertainty in the momentum (∆px) of the particle in its lowest energy state is a) ∆px = 0 b) ∆px = h/a c) ∆px = h/2a d) ∆px = h/2πa 8. The ground state electronic energy (Hartree) of a helium atom, neglecting the inter-electron repulsion, is a) -1.0 b) -0.5 c) -2.0 d) -4.0 9. A particle is confined to a one dimensional box of length 1 mm. If the length is changed by 10-9 m, the % change in the ground state energy is a) 2x10-4 b) 2x10-7 c) 2x10-2 d) 0 10. 10. The operation of the commutator [x, d/dx] on a function f(x) is equal to a) 0 b) f(x) c) –f(x) d) x df/dx 11. The lowest energy state of the (1s)2 (2s)1 (3s)1 configuration of Be is a) 1S0 b) 1D2 c) 3S1 d) 3P1 12. For hydrogen like atom with a nuclear charge Z, the energy of orbital with principal quantum number ‘n’ follows the relation a) En∞n2Z2 b) En∞ - Z2/n c) En∞ - Z/n d) En∞ - Z2/n2 13. The average value of the radius

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15. Using Huckel molecular orbital approximation, the two roots of secular equation of ethane are a) α + √2β, α-√2β b) α +β, β c) α +β, α - β d) α + 2β, α = 2β 16. The term symbol of a molecule with electronic configuration (1σg)2 (1σu)2 (2σg)2 (2σu)2 (1πu)1 (1πu)1 is a) 1 Σ+g b) 3 Σ-g c) 1 Σ+-g d) 3 Σ+g 17. In general ∫ψ *ψ dτ , what are the dimensios of ψ a) Unitless

b) m3

c) m-3/2

d) m+1/2

18. The highest occupied MO in N2 and O+2 respectively are (take xaxis as internuclear axis) a) σ2px, π*2py b) π2py, π*2pz * c) σ 2px, σ2px d) π*2py, π2pz 19. The number of microstates for d6 electronic configuration is a) 210 b) 14x63 c) 7x62 d) 28x63 20. The work function for metallic cesium is 2.14 eV. Calculate the kinetic energy and the speed of the electrons ejected by light of wavelength 700 nm. a) -3.7 b) 3.7 c) 3.7x1010 d) -3.7x1010 SECTION - B (10X2=20) Answer any TEN Questions 21. Calculate the average energy of an oscillator with frequency 10 Hz(S-1) at a temperature 1000 K. 22. Which of the following operators are linear?

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(i ) (ii)

Aˆϕ = λϕ ˆ ψ =ψ * B

(λ =constant)

(iii ) Cˆ ϕ =ϕ 2 (iv) (v )

ˆ ψ = dψ D dx 1 Eˆψ =

ψ

a.(i), (ii) and (v)

b. (iii) and (v) c. (i) and (iv) d. (i), (iv) and (v)

23. Determine the normalisation constant N, the wave function for the 2s orbital of a hydrogen atom is N(2-r/a0)e-r/2a0. 24. Calculate the density functional for 2pz orbital. 25. For particle in 1-D box, the reasonable guess is Ψ = Nx (L-x), Calculate the expected ground state average energy. 26. The work function for the platinum metal is 8.0x10-10J. Will a radiation of wavelength 200nm be able to cause photoelectric effect in it? If so, what will be the velocity of the electron ejected from the surface? (Given 10-10J=5042.7 cm-1, and mass of the electron=9.1x10-31kg, c=3x1010 cm s-1). 27. Which among the following is/are true? A wave function is given as =sin x. (i) acceptable (ii) normalized (ii) eigen function of momentum operator pxˆ a. All the above b. (i) and (iii) c. (i) d. (ii) 28. Verify by the use of determinantal wave function that for Li atom a configuration like 1 s3 is impossible. 29. Calculate the first ionisation potentials for Be on the basis of Slater’s rule.

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30. Determine the term symbols for L=2,S=1/2 31. Calculate the zero-point energy of a harmonic oscillator consisting of a particle of mass 5.16x10-26 kg and force constant 285 Nm-1. 32. An electron moving in a simple harmonic potential V = ½ Kx2 is subjected to a perturbation H = Ex where E is the strength of electric field which is applied in x-direction. Determine the effect of first order perturbation on the energy. SECTION – C (5X7=35) Answer any FIVE Questions 33. a) Calculate the average energy of an oscillator with frequency 10 Hz(S-1) at a temperature 1000 K. (2) b) Determine the normalization constant. (2) Take a trial function for Helium atom as, ψ = φ(1) φ(2) with φ(1) = (z’3/π)1/2 e-z’r1 Φ(2) = (z’3/π)1/2 e-z’r2 c) Calculate the Zero-point of a harmonic oscillator consisting of a particle of mass 5.16x10-26 kg and force constant 285 N m-1 (3) 34. a) For a harmonic oscillator of effective mass 2.88x10-25 kg, the difference in adjacent energy level is 3.17 zJ. Calculate the force constant of the oscillator. (3) b) Find the best value of Z’ (2) c) Calculate the energy per photon for radiation of wavelength 600 nm (red). (2) 35. a) The diameter of the Sun is 1.4x109m and the temperature of its surface is 5800 K. Assuming it a black body, estimate the energy loss per second by radiation from the Sun, given that the total radiant energy emitted per second by unit area of a black

( 2π body is

5

k 4B )

2

15c h

3

T 4,

(3)

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b) Calculate the first and second ionization potentials.

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36. a) Calculate the most probable distance of an electron in the 2p state of H-atom. (3) b) Planck’s law of distribution of energy in the black body radiation is given by the equation, 8π hv3 dv E (v) dv = . Where E (v) is the energy 3 exp(hv / k B T)-1 c associated with the radiation of frequency v. Show that (i) the total energy of all radiations from the black body is proportional to the fourth power of the temperature of the body (Stefan-Boltzmann law); and (ii) the wavelength corresponding to the maximum energy density is inversely proportional to the temperature (Wien’s law). (2+2) 37. a) Calculate the average value of x for the 1s electron in H-atom.(2) b) Make a reasonable guess at the ground state wave function for H-atom. Use variation method to calculate the energy and normalized wave function. Show that the function is an Eigen function of Hamiltonian operator. (3) c) Construct an anti symmetric wave function for Be atom as a linear combination of products of orbitals. (2) 38. a) Show that Y1,1(θ, φ) and Y2,0 (θ, φ) are orthogonal. b) Find out the radial function for 3s,3p orbitals.

(3.5) (3.5)

39. a) The speed of a certain proton is 0.45 Mm s-1. If the uncertainty in its momentum is to be reduced to 0.0100 per cent, what uncertainty in its location must be tolerated? (2.5) b) How many microstates exist for d2 configuration? (2) c) Show that there can be no term symbol like 3D0 (2.5) ***********

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