QM Expectation Values for a Particle in a 1D Potential Well
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Solution Summary
For a particle with a given wavefunction in a one-dimensional potential well, we compute the expectation values of its position x, as well as the expectation value of x^2.
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a. We have
1 = integral_-4^6(Psi^*(x) Psi(x) dx)
= integral_-4^1(A^2 (4 + x)^2 dx) + integral_1^6(A^2 (6 - x)^2 dx)
= A^2 [integral_-4^1((16 + 8x + x^2) dx) + integral_1^6((36 - 12x + x^2) dx)]
= A^2 {[16x + 4x^2 + 1/3 x^3]_{x=-4}^1 + [36x - 6x^2 + 1/3 x^3]_{x=1}^6]}
= A^2 [16 + 4 + 1/3 - (-64) - 4(16) - 1/3(-64) + 36(6) - 6(36) + 1/3(216) - 36 + 6 - 1/3]
= A^2 / 3 (48 + 12 + 1 ...
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