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Problem 1 A particle is represented (at time $t=0$) by the wave function $Psi(x, t) = egin{cases} A(a^2 - x^2) & -a le x le +a \ 0 & ext{otherwise} end{cases}$ (1) a Determine the normalization constant A. b What is the expectation value of x? c What is the expectation value of p? (Note that you cannot get it from $<p> = m frac{d<x>}{dt}$. Why not?) d Find the expectation value of $x^2$. e Find the expectation value of $p^2$. f Find the uncertainty in $x$, ($sigma_x$). g Find the uncertainty in $p$, ($sigma_p$). h Check that your results are consistent with the uncertainty principle. Problem 2 Consider a particle in a potential well, $V(x) = egin{cases} 0 & -L le x le +L \ infty & ext{otherwise} end{cases}$ (2) Derive the time independent wave function. Find the time-dependent wave function.

          Problem 1
A particle is represented (at time $t=0$) by the wave function
$Psi(x, t) = egin{cases} A(a^2 - x^2) & -a le x le +a \ 0 & 	ext{otherwise} end{cases}$ (1)
a Determine the normalization constant A.
b What is the expectation value of x?
c What is the expectation value of p? (Note that you cannot get it from $<p> = m frac{d<x>}{dt}$. Why not?)
d Find the expectation value of $x^2$.
e Find the expectation value of $p^2$.
f Find the uncertainty in $x$, ($sigma_x$).
g Find the uncertainty in $p$, ($sigma_p$).
h Check that your results are consistent with the uncertainty principle.
Problem 2
Consider a particle in a potential well,
$V(x) = egin{cases} 0 & -L le x le +L \ infty & 	ext{otherwise} end{cases}$ (2)
Derive the time independent wave function.
Find the time-dependent wave function.

$Problem 1 A particle is represented (at time t=0) by the wave function Psi(x, t) = egincases A(a^2 - x^2) -a le x le +a 0 extotherwise endcases (1) a Determine the normalization constant A. b What is the expectation value of x? c What is the expectation value of p? (Note that you cannot get it from <p> = m fracd<x>dt. Why not?) d Find the expectation value of x^2. e Find the expectation value of p^2. f Find the uncertainty in x, (sigmax). g Find the uncertainty in p, (sigmap). h Check that your results are consistent with the uncertainty principle. Problem 2 Consider a particle in a potential well, V(x) = egincases 0 -L le x le +L infty extotherwise endcases (2) Derive the time independent wave function. Find the time-dependent wave function.$

Added by Mar M.

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