I believe a,b, and c are correct. Double check and solve the rest please. A part

Question

I believe a,b, and c are correct. Double check and solve the rest please. A particle is represented by the following wave function: Use the normalization condition to find C. L=2/L Evaluate the probability to find the particle in an interval of width 0.010L at x = L/4 (that is, between x = 0.245L and x = 0.255L. No integral calculation is necessary for this calculation.)P=0.01 Evaluate the probability to find the particle between x = 0 and x = +L/4. P=0.375 Find the average value of x and rms of x: x = Show that the average value of x2 in the one-dimensional infinite potential energy well is L2(1/3-1/2n2 pi2) Use the result above to show that, for the infinite potential energy well, defining

Explanation / Answer

part (b):

P = ^2 x

P = C^2 (-2x/L + 1)^2 x

P = ((3/L))^2 (-2(L/4)/L + 1)^2 (0.010 L)

P = (3/L) (-1/2 + 1)^2 (0.010 L)

P = (3) (0.5)^2 (0.010)

P = 0.0075

c)

P = C^2 (2x/L + 1)^2 dx

integration from 0 to L/4

P = (3/L) (4x^3/3L^2 - 2x^2/L + x)

P = (3/L) (4(L/4)^3/3L^2 - 2(L/4)^2/L + (L/4))

P = (3/L) (L/48 - L/8 + L/4)

P = (3) (1/48 - 1/8 + 1/4)

P = 0.4375

d)

<x> = * x dx

= ^2 x dx

= C^2(2x/L + 1)^2 x dx + C^2 (-2x/L + 1)^2 x dx

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Finding C^2(2x/L + 1)^2 x dx :

(3/L)(2x/L + 1)^2 x dx = (3/L) (2x/L)^2 * x + 4x^2/L + x dx

= (3/L) (4x^4/(4L^2) + 4x^3/(3L) + x^2/2)

= (-3/L) (4(-L/2)^4/(4L^2) + 4(-L/2)^3/(3L) + (-L/2)^2/2)

= -L/16

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Finding C^2 (-2x/L + 1)^2 x dx :

(3/L)(-2x/L + 1)^2 x dx = (3/L) (2x/L)^2 * x - 4x^2/L + x dx

= (3/L) (4x^4/4L^2 - 4x^3/3L + x^2/2)

= (3/L) (4(L/2)^4/4L^2 - 4(L/2)^3/3L + (L/2)^2/2)

= L/16

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Therefore:

<x> = * x dx

<x> = ^2 x dx

<x> = C^2(2x/L + 1)^2 x dx + C^2 (-2x/L + 1)^2 x dx

<x> = (-L/16) + L/16

<x> = 0

--------------------------------------------------------

---------------------------------------------------------

<x2> = * x dx

= ^2 x dx

= C^2(2x/L + 1)^2 x2 dx + C^2 (-2x/L + 1)^2 x2 dx

-----------------------------------------------------------

Finding C^2(2x/L + 1)^2 x2 dx :

(3/L)(2x/L + 1)^2 x2 dx = (3/L) ((2x/L)^2 * x^2 + 4x^3/L + x^2) dx

= (3/L) (4x^5/(5L^2) + 4x^4/(4L) + x^3/3)

= (-3/L) (4(-L/2)^5/(5L^2) + 4(-L/2)^4/(4L) + (-L/2)^3/3)

= L^2/80

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Finding C^2 (-2x/L + 1)^2 x dx :

(3/L)(-2x/L + 1)^2 x2 dx = (3/L) ((2x/L)^2 * x^2 - 4x^3/L + x^2) dx

= (3/L) (4x^5/(5L^2) - 4x^4/(4L) + x^3/3)

= (3/L) (4(L/2)^5/(5L^2) - 4(L/2)^4/(4L) + (L/2)^3/3)

= L^2/80

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Therefore:

<x2> = * x^2 dx

<x2> = ^2 x^2 dx

<x2> = C^2(2x/L + 1)^2 x2 dx + C^2 (-2x/L + 1)^2 x2 dx

<x2> = L^2/80 + L^2/80

<x2> = L2/40

(<x2>) = L/40

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