Use the following equation statement to answer the question. The normalized wave functions for the infinite square-well potential are
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-The expectation value <x> for a particle in its second excited (n = 3) state in a box of length L is

Question

Use the following equation statement to answer the question. The normalized wave functions for the infinite square-well potential are
 You may find it useful to use

-The expectation value <x> for a particle in its second excited (n = 3)state in a box of length L is
A)L 
B)3L/4 
C)L/2 
D)L/4 
E)L/8

Quizplus · Accepted Answer

The expectation value of x in any state n of the infinite square-well potential is given as:

⟨x⟩n = ∫_0^L x|ψn(x)|^2 dx

For the second excited state, n = 3. 
Therefore, the wave function for this state is given by:

ψ3(x) = √(2/L)sin(3πx/L)

So,

⟨x⟩3 = ∫_0^L x|ψ3(x)|^2 dx
       = ∫_0^L x(2/L)sin^2(3πx/L) dx
       = L/2

Hence, the expectation value of x for a particle in its second excited state in a box of length L is L/2. Therefore, the correct answer is C) L/2.

Use the Following Equation Statement to Answer the Question