The one-dimensional harmonic oscillator has a potential V(x) = ka² /2. We Henote the solutions of the associated Schrödinger equation as (x), n = ), 1, 2, .... The energy of state Pn(x) is En = hw(n + 1/2) (1.73) vhere w = Vk/m. Now consider the two-dimensional harmonic oscillator which has a po- cential
The one-dimensional harmonic oscillator has a potential V(x) = ka² /2. We Henote the solutions of the associated Schrödinger equation as (x), n = ), 1, 2, .... The energy of state Pn(x) is En = hw(n + 1/2) (1.73) vhere w = Vk/m. Now consider the two-dimensional harmonic oscillator which has a po- cential
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![Problem 1.14
The one-dimensional harmonic oscillator has a potential V(x) = ka² /2. We
denote the solutions of the associated Schrödinger equation as n(x), n =
0, 1, 2, ... The energy of state „(x) is
En = hw(n + 1/2)
(1.73)
where w = V
Now consider the two-dimensional harmonic oscillator which has a po-
Vk/m.
tential
k
V (2, y) = (2² + y*).
(1.74)
1. Prove that Vmn(x, y)
for the two-dimensional harmonic oscillator and find the corresponding
energy. [Hint: Group the second x-derivative term together with the
ka? /2 part of the potential, and do the same for y. Be sure fully to
exploit the fact that m and n solve the Schrödinger equation for the
one-dimensional harmonic oscillator.]
Vm(x)vn(y) solves the Schrödinger equation
2. What is the ground state energy for the two-dimensional harmonic
ocillator?
3. What is the next lowest energy after the ground state, and which com-
binations of (m, n) yield this energy?
Problem 1.15](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8925e98c-405f-4083-81be-588b6c325eb8%2Fc177bb0d-36c4-4862-9f1e-7c132875e655%2Fecvh8kp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Problem 1.14
The one-dimensional harmonic oscillator has a potential V(x) = ka² /2. We
denote the solutions of the associated Schrödinger equation as n(x), n =
0, 1, 2, ... The energy of state „(x) is
En = hw(n + 1/2)
(1.73)
where w = V
Now consider the two-dimensional harmonic oscillator which has a po-
Vk/m.
tential
k
V (2, y) = (2² + y*).
(1.74)
1. Prove that Vmn(x, y)
for the two-dimensional harmonic oscillator and find the corresponding
energy. [Hint: Group the second x-derivative term together with the
ka? /2 part of the potential, and do the same for y. Be sure fully to
exploit the fact that m and n solve the Schrödinger equation for the
one-dimensional harmonic oscillator.]
Vm(x)vn(y) solves the Schrödinger equation
2. What is the ground state energy for the two-dimensional harmonic
ocillator?
3. What is the next lowest energy after the ground state, and which com-
binations of (m, n) yield this energy?
Problem 1.15
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