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3. [15] Consider a particle in a one-dimensional box with infinite potential walls at x<0 and x>L. Inside of the box, we have a potential given by: V(x) = Vo(L-x) 0 ? x ? L where Vo is a constant. In this question we will use first order perturbation theory to calculate the energy of the particle. a. [2] Sketch the potential inside of the box. b. [5] Using as the unperturbed system, the ‘conventional’ particle in a box where V(x)=0 inside the box, explicitly define H?, H?? and H?'. c. [5] What is the ground state energy of the system corrected to first order? d. [3] Consider the ground state wave function corrected to first order. Qualitatively describe how the ground state and first two excited states of the unperturbed system may combine to form the ground state wave function corrected to first order. Sketch how the ground state wave function corrected to first order might look. Hint: You can use integrals.com to integrate definite integrals: For example you can type: integrate sin(a*x) from 0 to Pi and it will give you the definite integral: Definite integral: ??? sin(ax) dx = (1 - cos(?a)) / a If you use integrals.com in this way, be sure to state this in your answer.

          3. [15] Consider a particle in a one-dimensional box with infinite potential walls at x<0 and x>L. Inside of the box, we have a potential given by:
V(x) = Vo(L-x) 0 ? x ? L
where Vo is a constant. In this question we will use first order perturbation theory to calculate the energy of the particle.
a. [2] Sketch the potential inside of the box.
b. [5] Using as the unperturbed system, the ‘conventional’ particle in a box where V(x)=0 inside the box, explicitly define H?, H?? and H?'.
c. [5] What is the ground state energy of the system corrected to first order?
d. [3] Consider the ground state wave function corrected to first order. Qualitatively describe how the ground state and first two excited states of the unperturbed system may combine to form the ground state wave function corrected to first order. Sketch how the ground state wave function corrected to first order might look.
Hint: You can use integrals.com to integrate definite integrals: For example you can type:
integrate sin(a*x) from 0 to Pi
and it will give you the definite integral:
Definite integral:
??? sin(ax) dx = (1 - cos(?a)) / a
If you use integrals.com in this way, be sure to state this in your answer.

3. [15] Consider a particle in a one-dimensional box with infinite potential walls at x<0 and x>L. Inside of the box, we have a potential given by:
V(x) = Vo(L-x) 0 ? x ? L
where Vo is a constant. In this question we will use first order perturbation theory to calculate the energy of the particle.
a. [2] Sketch the potential inside of the box.
b. [5] Using as the unperturbed system, the ‘conventional’ particle in a box where V(x)=0 inside the box, explicitly define H?, H?? and H?'.
c. [5] What is the ground state energy of the system corrected to first order?
d. [3] Consider the ground state wave function corrected to first order. Qualitatively describe how the ground state and first two excited states of the unperturbed system may combine to form the ground state wave function corrected to first order. Sketch how the ground state wave function corrected to first order might look.
Hint: You can use integrals.com to integrate definite integrals: For example you can type:
integrate sin(a*x) from 0 to Pi
and it will give you the definite integral:
Definite integral:
??? sin(ax) dx = (1 - cos(?a)) / a
If you use integrals.com in this way, be sure to state this in your answer.

Added by Corey F.

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