In quiz last week you considered a system described by a wave function of the formP(x) = N a (x-a) for o =dx $(x)* & ¢(x)in a system described by the wave function p(x)?O a (8) =5/a?Ob.(8) = x²/5O C.* (8) = (a + x)/2Od () = a/2Oe.° <8)= a²/4

The wave function is given byϕ(x)=N a(x-a) for 0≤x≤a =0x^=∫-∞+∞ dx ϕ*(x) x^ ϕ(x)…

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Question

In quiz last week you considered a system described by a wave function of the form
P(x) = N a (x-a) for o <xsa and p(x) = o otherwise.
In particular in problem (2) of Quiz 1 you found that the normalization constant N had the value N = 30/2/a5/2.
What is the value of the expectation value of the position of a particle
<8> =
dx $(x)* & ¢(x)
in a system described by the wave function p(x)?
O a (8) =5/a?
Ob.
(8) = x²/5
O C.
* (8) = (a + x)/2
Od () = a/2
Oe.
° <8)= a²/4

Expert Solution

Step 1

$T h e w a v e f u n c t i o n i s g i v e n b y ϕ (x) = N a (x - a) f o r 0 \leq x \leq a = 0 <\hat{x}> = \int_{- \infty}^{+ \infty} d x ϕ^{*} (x) \hat{x} ϕ (x) = \int_{- \infty}^{0} 0 d x + \int_{0}^{a} {[N a (x - a)]}^{2} x d x + \int_{a}^{\infty} 0 d x = N^{2} a^{2} \int_{0}^{a} {(x - a)}^{2} x d x = N^{2} a^{2} \int_{0}^{a} (x^{2} + a^{2} - 2 a x) x d x = N^{2} a^{2} \int_{0}^{a} (x^{3} d x + a^{2} x d x - 2 a x^{2} d x) = N^{2} a^{2} (\frac{x^{4}}{4} + \frac{a^{2} x^{2}}{2} - \frac{2 a x^{3}}{3})$

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In quiz last week you considered a system described by a wave function of the form P(x) = N a (x-a) for o