qmhw8 - 3. A particle of mass m interacts with a central...

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PHY4221: Homework 8 Due Nov. 6, 2008 1. The Hamiltonian of a nonlinear oscillator of mass m is given by, H = ~ ω ± a a + 1 2 + ga a aa where ω and g are constants, and a and a are annihilation and creation operators. (a) Let the energy eigenstates of the linear oscillator with g = 0 be | n i , show that | n i are also the energy eigenstates of the nonlinear oscillator for any values of g . Then find the energy eigenvalues of H . (b) The system described by the Hamiltonian H above is prepared initially in the state (assuming α is real): | ψ (0) i = 1 1 + α 2 ( | 0 i + α | n i ) . Discuss how the nonlinearity of the system affects the expectation value of position as time increases if n = 1 and n = 2. 2. A particle of mass m is trapped in the space a r b formed by the spherical walls of radii a and b . Solve all the energy eigenvalues and eigenfunctions of the system with zero orbital angular momentum.
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Unformatted text preview: 3. A particle of mass m interacts with a central potential defined by V ( r ) = gδ ( r-a ) where g and a are positive constants. Solve all the energy eigenfunctions of the system with zero orbital angular momentum. 4. The wave function of a particle is given by, ψ ( ~ r ) = A ( r + x ) e-αr where A is a normalization constant, and r = p x 2 + y 2 + z 2 is the radial distance. It is given that Y 00 = 1 √ 4 π , Y 10 = r 3 4 π cos θ, Y 11 =-r 3 8 π sin θe iφ ,Y 1 ,-1 =-Y * 11 . (a) Express the wave function in terms of spherical harmonics. (b) What is the expectation value of L 2 ? (c) If the z-component of the angular momentum is measured, what are the possible outcomes and the corresponding probabilities? 1...
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