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Fund Quantum Mechanics Lect & HW Solutions 72

Fund Quantum Mechanics Lect & HW Solutions 72 - as...

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54 CHAPTER 4. SINGLE-PARTICLE SYSTEMS 4.5.4 States that share the electron 4.5.5 Comparative energies of the states 4.5.6 Variational approximation of the ground state 4.5.6.1 Solution hione-a Question: The solution for the hydrogen molecular ion requires elaborate evaluations of inner product integrals and a computer evaluation of the state of lowest energy. As a much simpler example, you can try out the variational method on the one-dimensional case of a particle stuck inside a pipe, as discussed in chapter 3.5. Take the approximate wave function to be: ψ = ax ( x ) Find a from the normalization requirement that the total probability of finding the particle integrated over all possible x -positions is one. Then evaluate the energy ( E ) as ( ψ | H | ψ ) , where according to chapter 3.5.3, the Hamiltonian is
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Unformatted text preview: as a ψ | H | ψ A , where according to chapter 3.5.3, the Hamiltonian is H = − ¯ h 2 2 m ∂ 2 ∂x 2 Compare the ground state energy with the exact value, E 1 = ¯ h 2 π 2 / 2 mℓ 2 (Hints: i ℓ x ( ℓ − x ) d x = ℓ 3 / 6 and i ℓ x 2 ( ℓ − x ) 2 d x = ℓ 5 / 30) Answer: To satisfy the normalization requirement that the particle must be somewhere, you need a ψ | ψ A = 1, or substituting for ψ , 1 = a ax ( ℓ − x ) | ax ( ℓ − x ) A = | a | 2 a x ( ℓ − x ) | x ( ℓ − x ) A And by de±nition, chapter 2.3, the ±nal inner product is just the integral i ℓ x 2 ( ℓ − x ) 2 d x which is given to be ℓ 5 / 30. So you must have | a | 2 = 30 ℓ 5...
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