# Neutrino/schrodinger eq problem

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1. Apr 9, 2013

### jennyjones

Hey,

I'm a high school student from Europe and my final paper is on Neutrino oscillations.

I practiced some basic quantum states(qbit), but i find it much harder for this neutrino problem.

I translated it in hope that some of you could give me some pointers. I left some parts of the theory(not needed to solve eq's) out so hope it is still clear.

I ATTACHED IT!

I hope someone can help,

thanx

jenny
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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2. Apr 9, 2013

### Staff: Mentor

Hi Jenny!

Could you please post the Hamiltionian you are studying?

Also, what have you worked out up now ("The attempt at a solution")?

3. Apr 9, 2013

### jennyjones

Hey drClaude,

There is given that we start with an statonairy eigenstate of the Hamiltonian,

H|v1} = E1|v1}

I will try to scan in how i tried the problem

Last edited: Apr 9, 2013
4. Apr 9, 2013

### jennyjones

for a: this is what i have so far(attachment)

and then i would fill in a(0)=1

so,

1=a(t)*e^0=a(t)
a(t)=1

#### Attached Files:

• ###### ant.png
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Last edited: Apr 9, 2013
5. Apr 9, 2013

### Staff: Mentor

I see, you are not told what the Hamiltonian looks like!

Your start is good, although I don't see why you are considering that the wave function is a two-component vector.

To continue, you should use the initial condition for $a$, and probably also the time-independent Schrödinger equation in the quote above.

6. Apr 9, 2013

### jennyjones

Thanx! I'm not sure why i used the two component vector... but i see your point it;s unnecessary
as there is only one component

i don;t get tho what i'm suppost to do with the time-independent Schrödinger equation from the quote

7. Apr 9, 2013

### Staff: Mentor

What is the right-hand-side of the time-dependent Schrödinger equation?

8. Apr 9, 2013

### jennyjones

H|ψ(t)}?
is that equal to H|v1}

I don't get tho why i just can't say after i found a(t)=a_0 e^(-i(A/h_bar)t)
with the given condition condition a(0)=1

1=a*e^(0)=a

so a(t)=1

hmmm, maybe i have to take a better look at your hint before i keep asking questions

i will gve it an other go

9. Apr 9, 2013

### Staff: Mentor

$$i \hbar \frac{\partial | \psi(t) \rangle}{\partial t} = \hat{H} \psi(t) = \hat{H}( a(t) | \nu_1 \rangle) = a(t) \hat{H}| \nu_1 \rangle$$

You have two unknowns in there that need to be determined, $a_0$ and $A$. This is done using the equation I just wrote above and the condition that $a(0) = 1$.

10. Apr 9, 2013

### jennyjones

thanx DrClaude,

i'm going to give it another try!

11. Apr 10, 2013

### jennyjones

this is what i have so far

#### Attached Files:

• ###### Naamloos.png
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Last edited: Apr 10, 2013
12. Apr 10, 2013

### Staff: Mentor

Great, you have solved the first part:
$$a(t) =e^{-i E_1 t / \hbar}$$
You can move on to B.

13. Apr 10, 2013

### jennyjones

YEY! thanx

14. Apr 10, 2013

### jennyjones

is it ok if i used the norming conditing to solve b?

#### Attached Files:

• ###### Scan0003.jpg
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15. Apr 10, 2013

### jennyjones

no longer upside down

#### Attached Files:

• ###### Scan0003.jpg
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16. Apr 10, 2013

### Staff: Mentor

I think you have to.

In part B, it is not clear what $\alpha$ and $\beta$ are. I know what you mean, but I don't think it is a rigourous way of solving the problem. You should start by the definition of the probability of finding the system in a given state.

I will have a look at part C later.

17. Apr 10, 2013

### jennyjones

I see what you mean with alpha and beta

thanx!

18. Apr 10, 2013

### jennyjones

I think i solve c, or almost solved c

the thing i'm a bit worried about is that in my answer i get a -sin(alpha)

Btw i mistyped the assignment of c, the condition is |ψ(0)} = |v(μ)}
(instead of |ψ(0)} = |v(e)})

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