we derived the solution of Schrödinger's equation for a particle in a box in 1-D. We used the separation of variables, (x, t) = F(x)G(t), to get two separate differential equations: one for time and the other one for position. Using some constraints imposed by the fact that the system is real and physical, we get the solution G(t), by putting the constant into the F(x) term, of the form (11) where is an arbitrary constant and E is the total energy of the particle. On the other hand, the equation for the spatial component of the wave equation can be written as G(t) = e 0² F(x) ?х2 with 8² 8² + მე2 дуг F(x). (12) Now, in this problem, we want to extend this result into 2-D. The potential energy is -iEt/ħ =e V(x, y) = {0, 0≤x≤a and 0 ≤ y ≤ b otherwise, and the equation for the spatial component of the wave equation becomes 2m7² h J²X (x) əx² 2m7² ħ a²Y (y) dy² F(x,y) Similar to how we get two separate equations for time and position, we can use the separation of variables again by letting F(x, y) = X(x)Y(y). This gives a system of equations: -F(x, y). 2ma² ħ 2m3² h -X(x), (13) -Y (y), (14) (15) (16) a² + 3²= 7², where a and 3 are some other arbitrary constants. Using these results and the boundary conditions, we can derive the solution of 2- dimensional Schrödinger's equation for a particle in a 2-D box by treating it as a standing wave in 2-D. (17) (c) find F(x,y). Your answers should still have the mode numbers. [Hint: To find F(x, y), remember that there must be a constant multiplying with the x- and y-dependent terms such that ff (F(x, y))²dxdy = 1, and also remember that x and y are independent when evaluating the integral.] (d) Finally, find the complete wavefunction for each mode (x, y, t) = F(x, y)G(t). Write everything in terms of given variables and the mode numbers nx, ny. [Hint: use the expression for E found in the previous part.]

we derived the solution of Schrödinger's equation for a particle in a box in 1-D. We used the separation of variables, (x, t) = F(x)G(t), to get two separate differential equations: one for time and the other one for position. Using some constraints imposed by the fact that the system is real and physical, we get the solution G(t), by putting the constant into the F(x) term, of the form (11) where is an arbitrary constant and E is the total energy of the particle. On the other hand, the equation for the spatial component of the wave equation can be written as G(t) = e 0² F(x) ?х2 with 8² 8² + მე2 дуг F(x). (12) Now, in this problem, we want to extend this result into 2-D. The potential energy is -iEt/ħ =e V(x, y) = {0, 0≤x≤a and 0 ≤ y ≤ b otherwise, and the equation for the spatial component of the wave equation becomes 2m7² h J²X (x) əx² 2m7² ħ a²Y (y) dy² F(x,y) Similar to how we get two separate equations for time and position, we can use the separation of variables again by letting F(x, y) = X(x)Y(y). This gives a system of equations: -F(x, y). 2ma² ħ 2m3² h -X(x), (13) -Y (y), (14) (15) (16) a² + 3²= 7², where a and 3 are some other arbitrary constants. Using these results and the boundary conditions, we can derive the solution of 2- dimensional Schrödinger's equation for a particle in a 2-D box by treating it as a standing wave in 2-D. (17) (c) find F(x,y). Your answers should still have the mode numbers. [Hint: To find F(x, y), remember that there must be a constant multiplying with the x- and y-dependent terms such that ff (F(x, y))²dxdy = 1, and also remember that x and y are independent when evaluating the integral.] (d) Finally, find the complete wavefunction for each mode (x, y, t) = F(x, y)G(t). Write everything in terms of given variables and the mode numbers nx, ny. [Hint: use the expression for E found in the previous part.]

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help me with part d please, its not graded question

=
we derived the solution of Schrödinger's equation for a particle in a box in
1-D. We used the separation of variables, (x, t) F(x)G(t), to get two separate
differential equations: one for time and the other one for position. Using some
constraints imposed by the fact that the system is real and physical, we get the
solution G(t), by putting the constant into the F(xr) term, of the form
(11)
where is an arbitrary constant and E is the total energy of the particle. On
the other hand, the equation for the spatial component of the wave equation can be
written as
G(t) = e-i²t
0² F(x)
?х2
V(x, y) =
with
(12)
Now, in this problem, we want to extend this result into 2-D. The potential energy
is
=
= e
[0, 0≤x≤a and 0 ≤ y ≤ b
∞, otherwise,
and the equation for the spatial component of the wave equation becomes
2my²
F(x, y)
h
2² 8²
+
əx² Əy²
-iEt/ħ
2m7²
ħ
J²X (x)
əx²
0²Y (y)
дуг
-F(x).
-F(x, y).
Similar to how we get two separate equations for time and position, we can use the
separation of variables again by letting F(x,y) = X(x)Y(y). This gives a system of
equations:
2ma²
ħ
2m 8²
ħ
-X(x),
(13)
-Y (y),
(14)
(15)
(16)
a² + 3² = 7²,
where a and ß are some other arbitrary constants.
Using these results and the boundary conditions, we can derive the solution of 2-
dimensional Schrödinger's equation for a particle in a 2-D box by treating it as a
standing wave in 2-D.
(17)
(c)
find F(x, y). Your answers should still have the mode numbers. [Hint:
To find F(x, y), remember that there must be a constant multiplying with the x-
and y-dependent terms such that fo f(F (x, y))²dxdy = 1, and also remember
that x and y are independent when evaluating the integral.]
(d) Finally, find the complete wavefunction for each mode (x, y, t) = F(x, y)G(t).
Write everything in terms of given variables and the mode numbers nx, ny.
[Hint: use the expression for E found in the previous part.]

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