Parity Conservation: Determine E for P=-1

[Ed Quanta]

Ok,so check this situation out.

We have a one-dimensional box with walls at (-a/2,a/2). We know that the particle is in a state with energy probabilities

P(E1)=1/3, P(E2)=1/3, and P(E3)=1/3 while P(En)=0 for all n not equal to 1,2,3.

The parity is measured ideally and -1 is found. If some time later E is measured, what value is found? What is the answer if the original measurement found the parity to be 1?

I don't understand how if the parity of the state is measured ideally that -1 is found, since
we know Psi(x,0)= (square root of (2/3a))(cos(n(pi)x/a) + sin(2n(pi)x/a) + cos(3n(pi)x/a)

And when we take P, where P is the parity operator of our function, we note that only the sin function will become negative.

solving for <P> then, <P>= 1/3 + (-1/3) + 1/3=1/3, correct? This does not equal 1.


Anyway, given the initial state, now I am pretty sure that
Psi(x,t)= (square root of (2/3a))(cos(n(pi)x/a)e^-iE1t/hbar +
sin(2n(pi)x/a)e^-iE2t/hbar + cos(3n(pi)x/a)e^-iE3t/hbar)

How do I determine energy from this?
And then how will the energy change if the original parity happened to be -1?

 

[quantumdude]

And when we take P, where P is the parity operator of our function, we note that only the sin function will become negative.

Bingo. You measure a negative parity, so you must have found the particle in the only allowed basis state that has negative parity.

solving for <P> then, <P>= 1/3 + (-1/3) + 1/3=1/3, correct? This does not equal 1.

But they didn't say that you obtained an expectation value of -1, they said that you measured the particle with a parity of -1. That means that you found it in a negative parity eigenstate, and there's only one of those available, so...

Furthermore, if you find the parity to be 1, then you could be in either of the two even parity eigenstates.

 

[R0man]

The parity conservation law states that the parity of a physical system remains unchanged under certain transformations, such as reflection or inversion. In this case, we are given a one-dimensional box and the energy probabilities for a particle in that box. The parity is then measured and found to be -1. This means that the state of the system is not symmetric under reflection.

To determine the energy in this situation, we can use the equation E = n^2h^2/(8mL^2), where n is the quantum number, h is Planck's constant, m is the mass of the particle, and L is the length of the box. Since the parity is -1, we know that the quantum number n must be odd. This means that the only possible energy values are E1 = 9h^2/(8ma^2), E2 = 25h^2/(8ma^2), and E3 = 49h^2/(8ma^2).

If we were to measure the energy after finding the parity to be -1, we would find one of these three energy values with equal probability.

If the original measurement had found the parity to be 1, then the state of the system would be symmetric under reflection. This means that the quantum number n would be even, and the possible energy values would be E1 = 4h^2/(8ma^2), E2 = 16h^2/(8ma^2), and E3 = 36h^2/(8ma^2). In this case, if we were to measure the energy after finding the parity to be 1, we would find one of these three energy values with equal probability.

In summary, the measurement of parity does not directly give us the energy value, but it constrains the possible energy values based on the symmetry of the system. The actual energy value is determined through the use of the Schrödinger equation and the appropriate boundary conditions for the given system.