1. Consider the Schrödinger equation for a particle in the presence of the potential of the form V(x) = U(x) + (W(x) where U and W are real functions of x. What form does the conservation equation take? Hint: Use the Schrödinger equation and its complex conjugate with this potential and proceed in a manner like that of a free particle. The definitions of P(x, t) and j(x, t) are the same as for the free particle equations.
1. Consider the Schrödinger equation for a particle in the presence of the potential of the form V(x) = U(x) + (W(x) where U and W are real functions of x. What form does the conservation equation take? Hint: Use the Schrödinger equation and its complex conjugate with this potential and proceed in a manner like that of a free particle. The definitions of P(x, t) and j(x, t) are the same as for the free particle equations.
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Transcribed Image Text:1. Consider the Schrödinger equation for a particle in the presence of the potential
of the form
V(x) = U(x) + iW(x)
where U and W are real functions of x. What form does the conservation equation
take?
Hint: Use the Schrödinger equation and its complex conjugate with this potential
and proceed in a manner like that of a free particle. The definitions of
P(x, t) and j(x, t) are the same as for the free particle equations.
2. Show that for a one-dimensional square integrable wave function 4(x, t), we have
<p>
| dx j(x) =
m
where j(x) is the probability current.
Hint: Start from the definition of j(x) and integrate it by parts.
3. Consider the ground state wave function of the one-dimensional harmonic
ocillator given by
max
p(x) = A e
3.1 Find the value of the normalization constant A.
3.2 Calculate the expectation values of the momentum p and its square p? for this
wave function.
4. Consider the function (6) of the angular variable e, restricted to the interval
-n sOS n. If the wave functions satisfy the condition (n) = 4(-n), show
that the operator
hd
L =
ī de
has a real expectation value.
5. Solve the Schrödinger equation for a particle in a box with sides at x = -L and
x = L. Determine the eigenvalues and the normalized eigenfunctions.
Hint: After obtaining the general solution, separate it into odd and even solutions
and apply the boundary conditions to each separately. Then obtain odd energy
eigenyalues and eigenfunctions oand do the same for even functione
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