Homework Help is Here – Start Your Trial Now!
Science
Advanced Physics
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 G(t)=e-²t = e (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 ²F(x) дх2 with (12) Now, in this problem, we want to extend this result into 2-D. The potential energy is 8² 8² + əx² მყ2 -iEt/ħ 2m7² h V(x, y) ∞, otherwise, and the equation for the spatial component of the wave equation becomes 2m² F(x,y). h F(x,y) ²X (x) дх2 ²Y (y) dy² 1 -F(x). 0≤x≤ a and 0 ≤ y ≤ b 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 -X(x), 2m32 ħ (13) -Y(y), (14) (15) (16) a² +8² = 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) Then, 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 fof(F (x, y))²dxdy = 1, and also remember that x and y are independent when evaluating the integral.]

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 G(t)=e-²t = e (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 ²F(x) дх2 with (12) Now, in this problem, we want to extend this result into 2-D. The potential energy is 8² 8² + əx² მყ2 -iEt/ħ 2m7² h V(x, y) ∞, otherwise, and the equation for the spatial component of the wave equation becomes 2m² F(x,y). h F(x,y) ²X (x) дх2 ²Y (y) dy² 1 -F(x). 0≤x≤ a and 0 ≤ y ≤ b 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 -X(x), 2m32 ħ (13) -Y(y), (14) (15) (16) a² +8² = 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) Then, 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 fof(F (x, y))²dxdy = 1, and also remember that x and y are independent when evaluating the integral.]

icon
Related questions
Question

help me with part c please

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 G(t) = e-i²t = e-iEt/ħ (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 -F(x). (12) Now, in this problem, we want to extend this result into 2-D. The potential energy is V(x, y) = with 0² F(x) дх2 8² əx² + Jo, ∞, and the equation for the spatial component of the wave equation becomes 8² Jy² ) F(x, y) = 2m² h J²X (x) əx² 0≤x≤a and 0 ≤ y ≤ b otherwise, a²Y (y) dy² 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 h 2m² h -F(x, y). 2m 8² 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) Then, 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.]
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(x) term, of the form G(t) = e-i²t = e-iEt/ħ (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 -F(x). (12) Now, in this problem, we want to extend this result into 2-D. The potential energy is V(x, y) = with 0² F(x) дх2 8² əx² + Jo, ∞, and the equation for the spatial component of the wave equation becomes 8² Jy² ) F(x, y) = 2m² h J²X (x) əx² 0≤x≤a and 0 ≤ y ≤ b otherwise, a²Y (y) dy² 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 h 2m² h -F(x, y). 2m 8² 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) Then, 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.]
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 7 steps with 7 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer