Probability of measuring specific energy of particle (in a box)

[doorknob]

Homework Statement

Consider a particle-in-a-box problem, involving a
particle of mass m subject to a potential:
V(x) = +∞ for X≤0
V(x) = 0 for 0<X<L
V(x) = +∞ for X≥L

|ϕn> are the eigenstates of the Hamiltonian H with the corresponding eigenvalues:
En = (n^2)*(hbar^2)*(Pi^2) / 2mL^2

The state of the particle at the instant t = 0 is:
|Ψ(0)> = c1 |ϕ1> + c2 |ϕ2> + c3 |ϕ3> + c4 |ϕ4>

where ci are complex coefficients that satisfy Σ|ci|^2 = 1 (i.e., the state is normalized).

a) When the energy of the particle in the state |Ψ(0)> is measured what is the probability of
finding a value smaller than
3*(hbar^2)*(Pi^2) /mL^2?

b) What is the mean energy <H> and the uncertainty ΔE = [<H^2> − <H>^2] in the energy of the particle in the state |Ψ(0)> ?

c) Give the expression for the state vector |Ψ(t)> at time t. Do the results found in parts (a) and
(b) for t = 0 remain valid at an arbitrary time t? Make sure to show how you arrived at your conclusion.

d) The energy was measured at time t and the result
8*(hbar^2)*(Pi^2) /mL^2?
was found. After this measurement,
what is the state of the system? What is the result if the energy is measured again?

The Attempt at a Solution

a) the probability of obtaining an eigenvalue corresponding to the eigenstate |Ψ(t)> is:

|<ϕi|Ψ(t)>|^2

However, how do I compare the value to 3*(hbar^2)*(Pi^2) /mL^2? I do not know what the complex coefficients (c1,c2..c4) are.

b) Expectation energy <H> is defined as:
<H> = <Ψ|H|Ψ>

=> <H> = c1*E1<ϕ1|ϕ1> + c2*E2<ϕ2|ϕ2> + c3*E3<ϕ3|ϕ3> + c4*E4<ϕ4|ϕ4>
since normalized, => <ϕi|ϕi> = 1
So <H> = c1*E1 + c2*E2 + c3*E3 +c4*E4
But I do not know what the complex coefficients (c1,c2..c4) are.

c) don't know since have not completed part (a) or (b)

d) I would guess 3nd excited state (I'm assuming n=1 is the ground state)? Since if n=4, then 4^2 = 16.

16*(hbar^2)*(Pi^2) /2mL^2
reduces to 8*(hbar^2)*(Pi^2) /mL^2

After a measurement has been made, the system |Ψ> collapses to one of its eigenstates where it remains forever. If the energy were measured again, it should remain the same value as when you first made the measurement (since the system collapsed).

Any comments, feedback, criticism, would be helpful. I am more interested in the concept and understanding the material. Thank you in advance!

 

[vela]

Homework Statement

Consider a particle-in-a-box problem, involving a
particle of mass m subject to a potential:
V(x) = +∞ for X≤0
V(x) = 0 for 0<X<L
V(x) = +∞ for X≥L

|ϕn> are the eigenstates of the Hamiltonian H with the corresponding eigenvalues:
En = (n^2)*(hbar^2)*(Pi^2) / 2mL^2

The state of the particle at the instant t = 0 is:
|Ψ(0)> = c1 |ϕ1> + c2 |ϕ2> + c3 |ϕ3> + c4 |ϕ4>

where ci are complex coefficients that satisfy Σ|ci|^2 = 1 (i.e., the state is normalized).

a) When the energy of the particle in the state |Ψ(0)> is measured what is the probability of
finding a value smaller than
3*(hbar^2)*(Pi^2) /mL^2?

b) What is the mean energy <H> and the uncertainty ΔE = [<H^2> − <H>^2] in the energy of the particle in the state |Ψ(0)> ?

c) Give the expression for the state vector |Ψ(t)> at time t. Do the results found in parts (a) and
(b) for t = 0 remain valid at an arbitrary time t? Make sure to show how you arrived at your conclusion.

d) The energy was measured at time t and the result
8*(hbar^2)*(Pi^2) /mL^2?
was found. After this measurement,
what is the state of the system? What is the result if the energy is measured again?

The Attempt at a Solution

a) the probability of obtaining an eigenvalue corresponding to the eigenstate |Ψ(t)> is:

|<ϕi|Ψ(t)>|^2

However, how do I compare the value to 3*(hbar^2)*(Pi^2) /mL^2? I do not know what the complex coefficients (c1,c2..c4) are.

I'd expect your answer is supposed to be written in terms of the c_i's.

b) Expectation energy <H> is defined as:
<H> = <Ψ|H|Ψ>

=> <H> = c1*E1<ϕ1|ϕ1> + c2*E2<ϕ2|ϕ2> + c3*E3<ϕ3|ϕ3> + c4*E4<ϕ4|ϕ4>
since normalized, => <ϕi|ϕi> = 1
So <H> = c1*E1 + c2*E2 + c3*E3 +c4*E4
But I do not know what the complex coefficients (c1,c2..c4) are.

The c_i's are complex, so your expression for <H> will generally turn out to be a complex quantity. Does that make sense? You need to be a bit more careful in your calculations.

c) don't know since have not completed part (a) or (b)

d) I would guess 3nd excited state (I'm assuming n=1 is the ground state)? Since if n=4, then 4^2 = 16.

16*(hbar^2)*(Pi^2) /2mL^2
reduces to 8*(hbar^2)*(Pi^2) /mL^2

After a measurement has been made, the system |Ψ> collapses to one of its eigenstates where it remains forever. If the energy were measured again, it should remain the same value as when you first made the measurement (since the system collapsed).

That's right.

 

[doorknob]

Vela,
Thank you for your response. However, can I use the fact that I know:

Σ|ci|^2 = 1 (i.e., the state is normalized)

to obtain an expression without Ci? For example, when calculating mean energy <H>:
Expectation energy <H> is defined as:
<H> = <Ψ|H|Ψ>

*Note, (my mistake) my initial expression for <H> did not include the squared complex coefficients. It is now resolved to:
=> <H> = (c1^2)*E1<ϕ1|ϕ1> + (c2^2)*E2<ϕ2|ϕ2> + (c3^2)*E3<ϕ3|ϕ3> + (c4^2)*E4<ϕ4|ϕ4>

=> <H> = Σci*Ei*<ϕi|ϕi>

and since <ϕi|ϕi> = 1 we get:
Σ(|ci|^2)*ΣEi
= ΣEi

Is my line of reasoning correct?

 

[vela]

Vela,
Thank you for your response. However, can I use the fact that I know:

Σ|ci|^2 = 1 (i.e., the state is normalized)

to obtain an expression without Ci? For example, when calculating mean energy <H>:
Expectation energy <H> is defined as:
<H> = <Ψ|H|Ψ>

*Note, (my mistake) my initial expression for <H> did not include the squared complex coefficients. It is now resolved to:
=> <H> = (c1^2)*E1<ϕ1|ϕ1> + (c2^2)*E2<ϕ2|ϕ2> + (c3^2)*E3<ϕ3|ϕ3> + (c4^2)*E4<ϕ4|ϕ4>

That's better but still not quite right. I hope you realize that c_i²≠|c_i|².

=> <H> = Σci*Ei*<ϕi|ϕi>

and since <ϕi|ϕi> = 1 we get:
Σ(|ci|^2)*ΣEi
= ΣEi

Is my line of reasoning correct?

These remaining lines are all wrong.

 

[doorknob]

That's better but still not quite right. I hope you realize that c_i²≠|c_i|².

These remaining lines are all wrong.

Wow...okay, I guess I'll end up with an expression in terms of Ci's. Also, when you said "these remaining lines are all wrong" did you also mean that I was wrong to make the following assumption:

since normalized, => <ϕi|ϕi> = 1

 

[vela]

No, you're correct about the normalization.