How Do Neutrino Oscillations Relate to the Schrödinger Equation?

[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

 

[DrClaude]

Hi Jenny!

Could you please post the Hamiltionian you are studying?

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

 

[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

 

[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

 

[DrClaude]

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

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.

 

[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

 

[DrClaude]

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

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

 

[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

 

[DrClaude]

H|ψ(t)}?
is that equal to H|v1}

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

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

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 $a(0) = 1$.

 

[jennyjones]

thanx DrClaude,

i'm going to give it another try!

 

[jennyjones]

this is what i have so far

 

[DrClaude]

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

 

[jennyjones]

YEY! thanx

 

[jennyjones]

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

 

[jennyjones]

no longer upside down

 

[DrClaude]

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

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.

 

[jennyjones]

I see what you mean with alpha and beta

thanx!

 

[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)})