# Basic Two State System Question

• projektMayhem
In summary, a two state system can be described by linear superpositions of two stationary states, |E1> and |E2>, with corresponding energies E1 and E2. An observable Q has eigenstates |Q1> and |Q2>, with corresponding eigenvalues Q1 and Q2. The system is prepared in the state |E1> at t = 0 and is then measured by Alice and Bob at different times. The probability of measuring Q1 at t1 is (cos x)^2. At t2, the probability of measuring Q1 depends on whether or not Alice reported her result. If she did, the probability is (cos x)^4 + (sin x)^4 + 2(cos x

#### projektMayhem

A two state system is described by linear superpositions of two stationary states: |E1> and |E2>, with corresponding energies E1 and E2. An observable Q has the eigenstates |Q1> and |Q2>, corresponding to eigenvalues Q1 and Q2 (both real). All states are normalized.

Given: |Q1> = (cos x)|E1> + (sin x)|E2>

1) Express |Q2> as a superposition of |E1> and |E2> and determine the coefficients:

|Q2> = a|E1> + b|E2>

solution: Since Q is hermitian, the eigenstates are orthogonal ->
<Q1|Q2> = a (cos x) + b (sin x) = 0

=> a = -sin x; b = cos x (up to an overall phase factor - the minus sign could be switched)

2) The system is prepared at t = 0 to be in the state |E1>. It is then measured, first by Alice and then by Bob.

At t = t1, Alice measures Q. What is the probability that the measurement will yield Q1?
Here I will denote the state of the system as |X>

solution: |X(t)> = |E1> exp(-iE1 t / h)
P(Q1) = <Q1|X(t1)><X(t1)|Q1> = (cos x) exp (iE1 t1 /h) <E1|E1> cos (x) exp(-iE1 t1 / h) = (cos x) ^2

3) At t2 > t1 Bob measures Q. Again, determine the probability a measurement will yield Q1 - but consider the following two cases:

case 1) Alice reported that her result was Q1
case 2) Alice did not report the result of her measurement

Here is where I am confused... any help would be greatly appreciated :)

If my post has not met the guidelines, please notify me and i will modify where appropriate.
Thank you

I smell a cat! Someone call Mr. Schrodinger and ask him to pick it up.

Seriously, though - I think you need to consider what your initial state should be in each case, 3.1) and 3.2). You clearly have it for case 3.1), but you don't for 3.2), or rather, you don't know the outcome of Alice's observation, so you can't say which eigenstate it was left in; you only know probabilities for each possibility.

belliot, thanks for the response.

for case 1, is the probability unity?

i think the answer to my last question was no.

Keep in mind that the Q states are not the stationary states, so they evolve with time.

so, tell me if this is absurd:
for t1 < t < t2:

|X(t)> = (cos x)|E1>exp(-i E1 (t1 - t) / h) + (sin x)|E2>exp(-i E2 (t1 - t) / h)

then:

<Q1|X(t2)> = exp(-iE1 (t2 - t1)/h) [ (cos x)^2 + (sin x)^2 exp(-i(E2 - E1)(t2 - t1)/h)

so, P(Q1) = (cos x)^4 + (sin x)^4 + 2(cos x)^2 (sin x)^2 cos[(E2 - E1)(t2-t1)/h]

This is strictly positive, and is <= 1 - but I'm not sure it's correct.

Looks good to me ...

I have to admit that I'm pretty rusty at good ol' QM, but FWIW, I don't see any mistakes. I'm sure someone else will point them out if there are any, however.

## 1. What is a basic two state system?

A basic two state system is a physical system that can exist in one of two distinct states at a given time. It is often used in physics and chemistry to model systems such as spin particles or electronic energy levels.

## 2. How does a basic two state system work?

A basic two state system works by transitioning between two distinct states, often referred to as "on" and "off". This transition can be triggered by an external stimulus or occur spontaneously due to the system's internal energy levels.

## 3. What are some examples of basic two state systems?

Some examples of basic two state systems include the spin of an electron, the polarization of light, and the magnetic orientation of a particle.

## 4. How is a basic two state system different from a complex system?

A basic two state system is a simplified model that only has two possible states, while a complex system can have multiple states and interactions between them. Basic two state systems are often used to understand the fundamental principles of more complex systems.

## 5. What is the significance of studying basic two state systems?

Studying basic two state systems allows us to better understand the behavior of more complex systems that can be broken down into simpler components. It also has practical applications in fields such as quantum computing and materials science.