Problem Set 4 Solution

Dx 11 6 continued substituting the expectation value

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Unformatted text preview: ψ 0 ( x ) , € energy eigenvalue must be given by the € E0 = € € 1 hν 0 . 2 6 . For the ground state of the harmonic oscillator, determine the expectation value of the potential energy and also determine the expectation value of the kinetic energy. Then verify that the s um of the k inetic and p otential energy expectation values is the total energy. Potential Energy Expectation Value For a normalized wavefunction, the expectation value V of potential energy is V = € ∞ * ∫−∞ ψ0 ( x) Vˆ ψ0 ( x) dx . ˆ Substituting the form of the wavefunction and using that V = 1 2 k x2, € = 1 2 ȹ α ȹ1 / 2 k ȹ ȹ ȹ π Ⱥ € ∫−∞ e−α x = V 1 2 ȹ α ȹ1 / 2 k ȹ ȹ ȹ π Ⱥ ∫−∞ x 2 e−α x ∞ 2 /2 x 2 e−α x ∞ 2 2 /2 dx dx . From integral tables, € ∞ 2 − bx 2 ∫ xe 0 1 1 ȹ π ȹ 2 dx = ȹ ȹ . 4b ȹ b Ⱥ Since this integral is defined only from 0 to ∞ , we have € ∞ ∫ ∞ 2 x 2 − bx dx = 2 €e ∫ 1 1 ȹ π ȹ 2 ȹ ȹ . 2b ȹ b Ⱥ 2 x 2 e− bx dx = 0 −∞ Substituting this result into the expression for V , € € V = 1/ 2 k ȹ α ȹ ȹ ȹ 2 ȹ π Ⱥ 1/ 2 k ȹ α ȹ...
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