Hamiltonian and the operators themselves demonstrate the following properties

# Hamiltonian and the operators themselves demonstrate

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Hamiltonian, and the operators themselves, demonstrate the following properties of the creation and destruction operators: (a) ˆ h = 1 2 ˆ a ˆ a + ˆ a ˆ a ( ) = ˆ a ˆ a + 1 2 = ˆ a ˆ a 1 2 (b) 0 ˆ 0 = = v a ψ (c) 1 0 ˆ = = v v a ψ ψ (d) 0 1 ˆ = = v v a ψ ψ 3. Consider a particle of mass m that is trapped in a potential that might be described as a half-harmonic oscillator, with a dash of particle in a box: V x ( ) = 1 2 kx 2 x > 0 x 0 a) Describe the boundary conditions b) Obtain the eigenvalues and eigenfunctions, showing your logic clearly

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4. Exploring the eigenvalues and eigenfunctions of the harmonic oscillator: (a) Using the recurrence relations that we will derive in class or otherwise, obtain polynomials proportional to the Hermite polynomials for v=0, 1, 2, 3, and 4. (b) For the polynomials you obtained in part (a), work out the value of their nodes. This is pretty easy up to v=3, and even v=4 can be evaluated without great pain. Mark the positions of the nodes on the energy level corresponding to these eigenstates for the dimensionless oscillator Hamiltonian. 5. Consider the following model of a vibrational potential energy ( E < V 0 ): x < 0: V ( x ) = 0 x a : V ( x ) = 0 x > a : V ( x ) = V 0 (a) Write the boundary conditions that must be satisfied by the wavefunction at the points where the regions join, and as x becomes very large. (b) Write the general form of the wavefunction in regions I, II and III. (c) Show, using the boundary conditions, that allowed values of the energy must satisfy the following equation: 2 2 0 8 tan mE E a V E h π = (d) Adjust the model to fit the H 2 molecule. Here is some information that will be helpful. Its bond-length is 74 pm, and its well-depth is 7.6 × 10 -19 J. Also note that the mass m you should choose to describe the vibrating molecule mass should be the reduced mass of H 2 . Numerically find the vibrational eigenvalues that are bound.
• Summer '16
• Alistair Sinclair
• 10%, 19 J, 2070 cm, 2886 cm

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