Question

A particle of mass m is confined to the region x ∈ [0, L]. The wavefunction of the particle (at t = 0) is given by ψ(x) = A[sin(πx/L) + sin(2πx/L)] (this is the wavefunction from x = 0 to x = L, and ψ = 0 outside this region). (a) Write down the normalization condition for the wavefunction (this would let us solve for A). (b) Write down an expression for the probability that you will find the particle between x = 0 and x = L/2. (c) Plot ψ(x) from x = 0 to x = L. Based on your plot, is your answer to part (b) less than 50%, greater than 50%, or equal to 50%? Explain. (d) If we measured the energy of the state, explain what the possibilities are for the energy, and the probability that you'd see that value.

          A particle of mass m is confined to the region x ∈ [0, L]. The wavefunction of the particle (at t = 0) is given by ψ(x) = A[sin(πx/L) + sin(2πx/L)] (this is the wavefunction from x = 0 to x = L, and ψ = 0 outside this region).

(a) Write down the normalization condition for the wavefunction (this would let us solve for A).
(b) Write down an expression for the probability that you will find the particle between x = 0 and x = L/2.
(c) Plot ψ(x) from x = 0 to x = L. Based on your plot, is your answer to part (b) less than 50%, greater than 50%, or equal to 50%? Explain.
(d) If we measured the energy of the state, explain what the possibilities are for the energy, and the probability that you'd see that value.

Added by Philip C.

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