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PHY4221Fall05_Test1

PHY4221Fall05_Test1 - | Ψ(0 i =   1 ...

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PHY4221 Quantum Mechanics I Fall term of 2005 Test No.1 Answer ALL questions. Time: 2 1 2 hours 1. (10 points) An anti-Hermitian operator is equal to minus its Hermitian conjugate: ˆ Q = - ˆ Q . (a) (4 points) Show that the expectation value of an anti-Hermitian operator is imaginary. (b) (6 points) Show that the commutator of two Hermitian operators is anti- Hermitian. How about the commutator of two anti-Hermitian operators. 2. (25 points) Consider the potential V ( x ) = - ¯ h 2 a 2 m sech 2 ( ax ) , where a is a positive constant. Check that this potential has the ground state Ψ 0 ( x ) = A sech( ax ) , and find its energy. Normalise Ψ 0 ( x ) and sketch the wave function. 3. (30 points) The Hamiltonian of a certain three-level system is represented by the matrix H = a 0 b 0 c 0 b 0 a where a , b and c are real numbers. (a) (15 points) Find the normalised eigenstates and eigenvalues of the Hamiltonian H . (b) (10 points) If the system starts out in the state
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Unformatted text preview: | Ψ(0) i =    1    , then what is | Ψ( t ) i ? (c) (5 points) Compute the expectation value of H with respect to the state | Ψ( t ) i . 4. (35 points) The Hamiltonian operator ˆ H of a particle in a “double Dirac delta-function potential well” is given by ˆ H = ˆ p 2 2 m-V { δ (ˆ x + a ) + δ (ˆ x-a ) } where a is a real constant. (a) (12 points) Derive the transcendental equation whose roots give the eigenvalues of the bound states of even parity of ˆ H . (b) (12 points) Derive the transcendental equation whose roots give the eigenvalues of the bound states of odd parity of ˆ H . (c) (11 points) Show that there are altogether two bound states, one of even parity and one of odd parity. Sketch the two bound states....
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PHY4221Fall05_Test1 - | Ψ(0 i =   1 ...

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