5. A free particle has the following wave function at t = 0: V(x,0) = Ne-a|x| = [Ne-ª* x>0 Near x < 0 where N and a are real, positive constants. a) Normalize this wave function. b) Determine Ax by finding (x) and (x²). You can simplify the work by using symmetry (for example, if the wave function is symmetric around a = 0 what do you expect (x) to be?). The following integral could be useful: 1. me -bx dx = m! fm+1
5. A free particle has the following wave function at t = 0: V(x,0) = Ne-a|x| = [Ne-ª* x>0 Near x < 0 where N and a are real, positive constants. a) Normalize this wave function. b) Determine Ax by finding (x) and (x²). You can simplify the work by using symmetry (for example, if the wave function is symmetric around a = 0 what do you expect (x) to be?). The following integral could be useful: 1. me -bx dx = m! fm+1
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![5. A free particle has the following wave function at t = 0:
V(x, 0) = Ne- -a|x|
=
So
0
Ne-ax
Neax
where N and a are real, positive constants.
a) Normalize this wave function.
b) Determine Ax by finding (x) and (x²). You can simplify the work by using symmetry (for example, if the wave function
is symmetric around x = 0 what do you expect (x) to be?). The following integral could be useful:
-bx dx
xme
=
m!
fm+1
x > 0
x < 0
c) What is the expectation value of the momentum, (p)?
d) We can represent this wave function at t = 0 as a superposition (linear combination) of complex exponentials:
(x,0) = Ne-a|x| =[ A(k)etkx dk
-∞
Determine the amplitude function A(k).
e) Find an expression for (x, t). You can leave your answer in the form of an integral over k but make sure all dependence
on k is made explicit.
Optional: this choice of wavefunction V(x, 0) is really just for practice with calculating some expectation values, normal-
ization, etc, but it's actually a BAD choice for a wavefunction. Why is it a bad choice? Why might it fail at being a
realistic wavefuction?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F897405f4-fde2-48d6-b65b-90f81ac08b86%2F04a264d8-e22b-457b-a973-0feca557f398%2Flfws70j_processed.png&w=3840&q=75)
Transcribed Image Text:5. A free particle has the following wave function at t = 0:
V(x, 0) = Ne- -a|x|
=
So
0
Ne-ax
Neax
where N and a are real, positive constants.
a) Normalize this wave function.
b) Determine Ax by finding (x) and (x²). You can simplify the work by using symmetry (for example, if the wave function
is symmetric around x = 0 what do you expect (x) to be?). The following integral could be useful:
-bx dx
xme
=
m!
fm+1
x > 0
x < 0
c) What is the expectation value of the momentum, (p)?
d) We can represent this wave function at t = 0 as a superposition (linear combination) of complex exponentials:
(x,0) = Ne-a|x| =[ A(k)etkx dk
-∞
Determine the amplitude function A(k).
e) Find an expression for (x, t). You can leave your answer in the form of an integral over k but make sure all dependence
on k is made explicit.
Optional: this choice of wavefunction V(x, 0) is really just for practice with calculating some expectation values, normal-
ization, etc, but it's actually a BAD choice for a wavefunction. Why is it a bad choice? Why might it fail at being a
realistic wavefuction?
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