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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
G(t)=
(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
=e-i²t
² F(x)
əx²
V(x, y):
with
(12)
Now, in this problem, we want to extend this result into 2-D. The potential energy
is
2² 8²
+
əx² дуг
Jo, 0≤x≤a and 0 ≤ y ≤ b
[∞, otherwise,
and the equation for the spatial component of the wave equation becomes
-iEt/ħ
= e
²X (x)
əx²
2m² F(x).
ħ
0²Y (y)
Əy²
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:
2m7²
h
2ma²
ħ
2m3²
h
-F(x,y).
-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)
(a) Using 15, 16, and the boundary conditions (from the potential energy, the parti-
cle is confined in between 0 to a in the x-direction and 0 to b in the y-direction),
find the possible values of a and 3. Write your answers in terms of the parti-
cle's mass m and other given variables. You should also have the mode numbers
in each direction, na and ny, in your answers. [Hint: the boundary conditions
in the x-direction and y-direction can be used on X(x) and Y(y) separately,
respectively. The possible values of a and B then contain independent n values,
say na and ny, each can be any positive integer representing the modes in each
direction.]](https://content.bartleby.com/qna-images/question/335d6cef-e110-4df1-b3c9-cf74ce65a19c/267295f4-b378-4a65-8c41-846841386e4a/eajbuq1_processed.jpeg)
Transcribed Image Text:=
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
G(t)=
(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
=e-i²t
² F(x)
əx²
V(x, y):
with
(12)
Now, in this problem, we want to extend this result into 2-D. The potential energy
is
2² 8²
+
əx² дуг
Jo, 0≤x≤a and 0 ≤ y ≤ b
[∞, otherwise,
and the equation for the spatial component of the wave equation becomes
-iEt/ħ
= e
²X (x)
əx²
2m² F(x).
ħ
0²Y (y)
Əy²
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:
2m7²
h
2ma²
ħ
2m3²
h
-F(x,y).
-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)
(a) Using 15, 16, and the boundary conditions (from the potential energy, the parti-
cle is confined in between 0 to a in the x-direction and 0 to b in the y-direction),
find the possible values of a and 3. Write your answers in terms of the parti-
cle's mass m and other given variables. You should also have the mode numbers
in each direction, na and ny, in your answers. [Hint: the boundary conditions
in the x-direction and y-direction can be used on X(x) and Y(y) separately,
respectively. The possible values of a and B then contain independent n values,
say na and ny, each can be any positive integer representing the modes in each
direction.]
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