# #psi_A(x,0) = sqrt(1/6)phi_0(x) + sqrt(1/3)phi_1(x) + sqrt(1/2)phi_2(x)#? More questions

##
(d) What energy values will be observed as a result of a single measurement at t=0 and with what probabilities? How do these probabilities change with time?

(c) What is the probability of measuring the energy to equal #hatE# as a result of a

single measurement at t=0? At a later time t=t1?

(e) The energy of the particle is found to be E2 as a result of a single measurement

at t=t1. Write down the wave function ψA(x,t) which describes the state of the

particle for t> t1. What energy values will be observed and with what probabilities

at a time t2 > t1?

(f) Construct another normalized wave function ψB(x, 0) which is linearly independent of ψA(x, 0), but yields the same value of #hatE# as well as the same set of measured energies with the same probabilities.

(g) Construct another normalized wave function ψC(x, 0) which is linearly independent

of ψA(x, 0), yields the same value of (Eˆ), but allows a different set of measured

energies (which may include some but not all of E0, E1 and E2, plus others).

(d) What energy values will be observed as a result of a single measurement at t=0 and with what probabilities? How do these probabilities change with time?

(c) What is the probability of measuring the energy to equal

single measurement at t=0? At a later time t=t1?

(e) The energy of the particle is found to be E2 as a result of a single measurement

at t=t1. Write down the wave function ψA(x,t) which describes the state of the

particle for t> t1. What energy values will be observed and with what probabilities

at a time t2 > t1?

(f) Construct another normalized wave function ψB(x, 0) which is linearly independent of ψA(x, 0), but yields the same value of

(g) Construct another normalized wave function ψC(x, 0) which is linearly independent

of ψA(x, 0), yields the same value of (Eˆ), but allows a different set of measured

energies (which may include some but not all of E0, E1 and E2, plus others).

##### 1 Answer

#### Answer:

See below:

#### Explanation:

**Disclaimer** - I am assuming that

(d) The possible results of energy measurements are

These probabilities are independent of time (as time evolves, each piece picks up a phase factor - the probability, which is given by the modulus squared of the coefficients - do not change as a result.

(c) The expectation value is

Indeed,

(e)Immediately after the measurement that yields

At

The only possible value an energy measurement will yield on this state is

(f) The probabilities depend on the squared modulus of the coefficients - so

will work (there are infinitely many possible solutions). Note that since the probabilities have not changed, the energy expectation value will automatically be the same as

(g) Since

So a possible wavefunction (again, one of infinitely many possibilities) is