Chemistry 53 Practice Final Exam Fall Solutions x e ax dx π a 3/ ; π sin 3 xdx 4 3 π cos nx dx π; sin θ cos θ + K x n e ax dx n! a n+ ; r r r r ˆL h r ˆL z h i φ ˆL x i hsin φ + cot θ cos φ θ φ ) ˆLy i hcos φ cot θ sin φ θ φ ) ) / ) / h µω h ˆx [â + â ] ˆp i [â â ] µω π sin 5 x dx 6 5 All questions are worth 5 points.
. Briefly explain or define each of the following: 5 points each) a. The generalized momentum conjugate to the coordinate q i Let L{q i }, { q i }, t) be the classical Lagrangian for some physical system. Then the generalized momentum conjugate to the coordinate q i is defined by p i L q i. b. A functional A functional is a unique mapping between a set of functions called the domain to a set of numbers called the range. c. The null operator The operator Ô is the null operator if for any function f Ôf. d. The spherical harmonics The spherical harmonics {Y l,m θ, φ)} are the complete set of simultaneous eigenfunctions of the ˆL ˆL z operators. e. The Kronecker delta The Kronecker delta is defined by δ mn { m n m n
. Determine the uncertainty relations between the components of the angular momentum vector, L L x î+l y ĵ+l zˆk the components of the coordinate vector r xî+yĵ+zˆk. L x x [ˆL x, x] [y z z y ], x Then [ˆL x, x] h i L x x i.e. the x-component of L x can be simultaneously measured. Similarly, L y y L z z. Next [ˆL x, y] h i [y z z y, y ] h i [z y, y ] i hz L x y h z. Next [ˆL x, z] h i [y z z y, z ] h i [y z, z ] i hy L x z h y. Similarly L y x h z L y z h x L z x h y L z y h x 3
3. By constructing adjoints of the following operators, determine which are Hermitian. 5 points each) d a. dx Integrate by parts ) φ d dx φ dφ dx φ dx u φ dv dφ dx dx v φ du dφ dx dx ) φ d dx φ φ φ D φ d dx φ dx φ d dx φ dx Not Hermitian. b. i d dx φ i d dx ) d d dx dx ) φ dφ i dx φ dx φ i d dx φ dx Then i d ) i d dx dx Hermitian c. ˆLx ˆLy + ˆL y ˆLx ˆL x ˆLy + ˆL y ˆLx ) ˆL y ˆL x + ˆL x ˆL y Hermitian ˆL y ˆLx + ˆL x ˆLy 4
d. ˆLx ˆLy ˆL y ˆLx ˆL x ˆLy ˆL y ˆLx ) ˆL y ˆL x ˆL x ˆL y Not Hermitian ˆL y ˆLx ˆL x ˆLy e. i ˆL x or Not Hermitian φ iˆl x ) φ i φ ˆL x ) φ i φ ˆL x φ iˆl x ) iˆl x 5
4. The energy levels wavefunctions for a three-dimensional isotropic harmonic oscillator of reduced mass µ natural frequency ω can be solved in spherical polar coordinates where the potential energy takes the form V r) / µω r. In spherical polar coordinates, the radial wave equation takes the usual form [ h d dr r d dr h ll + ) + µr To solve this equation we introduce the transformations E µω r ) ] R nl r). α µω h ɛ Ē hω ξ αr R nl r) r l e αr / ur) resulting in the differential equation for ur) ξu ξ) + [ l + 3 ) ] + ξ u ξ) + [ l + 3 ) ] ɛ uξ). Use the power series method to solve the differential equation for uξ), truncate the recursion relation to obtain an expression for the energy levels of the system in terms of a principal quantum number n the angular momentum quantum number l. uξ) a n ξ n u ξ) a n nξ n u ξ) a n nn )ξ n n n n a n nn )ξ n + n In the first two sums, set a k+ kk + )ξ k + k or k l + 3 ) a n nξ n + n n a n nξ n + k n n k + l + 3 ) a k+ k + )ξ k + k k a k kξ k + [ l + 3 ) ] ɛ a n ξ n n [ l + 3 ) ] ɛ a k ξ k k { a k+ k + ) k + l + 3 ) [ + a k l + 3 ) ]} + k ɛ ξ k 6
giving the recursion relation a k+ a k Truncate at k n,,,..., [ ɛ k + ) ) ] k ) l + 3 k + l + 3 ɛ n n + l + 3 or E n hωɛ n n + l + 3 ) hω 7
5. At time t a particle of mass m in a one-dimension box of length L is in a non-stationary state represented by the normalized wavefunction ψx) x L ) 4 4 / L L 4 < x L L < x L Recalling that the normalized eigenfunctions eigenvalues of the Hamiltonian operator for a one-dimensional particle in a box are given by respectively ) / φ n x) sin nπx L L E n n π h ml if the energy of the particle is measured at time t calculate what values can be obtained with what probabilities? ψx) c n φ n x) L/ L/4 L n c n ψx)φ n x) dx ) / 4 / L/ sin L L) nπx L/4 L dx L y nπx L dx L nπ dy nπ/ c n L sin y dy L nπ nπ/4 cos nπ nπ 4 cos nπ ) Then we can measure E n with probability P E n ) 8 cos nπ n π 4 cos nπ L/ L/4 ) sin nπx L dx 8
6. An electron in a hydrogen atom occupies a 3d x y-orbital with associated normalized wavefunction ψ 3dx y r, θ, φ) ) 7/ r e r/3a sin θ cos φ 8π) / a where a is the Bohr radius. Calculate the expectation value of /r for the particle. r 8 π)a 7 π π dr dφ dθr sin θr e r/3a sin θ cos φ r r e r/3a sin θ cos φ 8 π)a 7 dr r 4 e r/3a π π dθ sin 5 θ dφ cos φ ) 5 3a 6π 4! 8 π)a 7 5 3 5a 9
7. A classical particle of mass m confined to move on the perimeter of a ring of radius R is subjected to a conservative force with associated potential energy function V θ) A cos θ where the constant A has units of energy θ is the angle in plane polar coordinates such that the connection between Cartesian coordinate plane polar coordinates is given by x R cos θ y R sin θ. Construct the classical Hamiltonian for the system in plane polar coordinates use Poisson s equation to derive an expression for dp θ /dt where p θ is the generalized momentum conjugate to the coordinate θ. L mẋ + ẏ ) V ẋ R θ sin θ ẏ R θ cos θ L mr θ A cos θ p θ L θ mr θ H θp θ L p mr θ + A cos θ θ mr + A cos θ dp θ dt {p θ, H} pb p θ H p θ H θ p θ p θ θ H θ A sin θ
8. The quantum Hamiltonian operator for the system described in problem 7 is given by Ĥ h d + A cos θ, mr dθ the quantum operator for the angular momentum associated with the coordinate θ is given by ˆp θ h d i dθ. Use the Heisenberg equation to derive an expression for d p θ /dt where the expectation value is taken with respect to any wavefunction representing the state of the system. Compare your result with problem 7. d p θ dt i h [ˆp θ, Ĥ] [ h/i)d/dθ, A cos θ] A sin θ i h The expectation value obeys the classical expression derived in part a.