Sequences of measurements in quantum mechanics

[ellenb899]

Homework Statement

Hi,

Whilst doing 1/2 spin measurement questions, a certain question typically repeats about the probability of finding a negative/positive outcome of an energy measurement of a quantum state.
So question is usually given, say wavefunction in
Sz fundamental basis |1> = (1) h/2 and |2> = (0) -h/2
(0) (1)
And wavefunction lets take (1 |1> + 3 |2>) can be normalized to unity aswell.

There is 8 probable questions(for t=0)- If measurement of state comes before energy measurement. If energy measurement comes before measurement of state. If one is positive and question asks for positive probability after calculation, if one is negative and question asks for positive after, if one is negative and question asks for negative after, and if one is positive and question asks for negative after calculation. For either of the 2 former questions.
Example :
-A measurement of energy takes place and gives positive value. Immediately after a measurement of Sx takes place. What is the probability that a negative value will result as an outcome?
-A measurement of Sy takes place and gives a negative value. Now, a measurement of energy takes place. What is the probability that a negative value will result as an outcome?

Relevant Equations

|a|^2. |b|^2.

ATTEMPT AT SOLUTION: I understand if looking for positive this will be +hwo/2 (hbar) for Sz so must find |a|^2. and if looking for negative this will be -hwo/2 (hbar) so must find |b|^2. If asked to find say Sx and original question in Sz, we must find new eigenstates associated with this state and recalculate.
Is there anything else I am forgetting? Or any pointers I could be given to help me do these questions with a bit more ease? I am fine with the math, its just I may need to write a bit of explanation beside it that I will need help with.

Thanks

 

[PeroK]

What energy measurement are you considering here?

 

[ellenb899]

probability of the wavefunction i have given

 

[PeroK]

probability of the wavefunction i have given

That doesn't define an energy measurement.

 

[ellenb899]

It is arbitrary. The question just asks for the probability of finding a negative or positive result for an energy measurement of the observable.

 

[PeroK]

It is arbitrary.

If you don't know what specific energy you are measuring, the how can you say what values you likely to get?

 

[ellenb899]

The wavefunction I provided. values 1 and 3. Or my mistake perhaps use 1/square root 10 and 3/square root 10 so they can be normalized to unity

 

[hutchphd]

What is the energy operator (the Hamiltonian) for the system? Is there a magnetic field? Which direction?

 

[ellenb899]

Yes magnetic field 1/2 spin in Sz state. z direction. Hamiltonian is qe/meSz which reduces to hwo/2 (1 0
0 -1) or H = woSz where wo = ebo/2me

 

[Chandra Prayaga]

Is this question still running?
I believe things would be clearer if standard symbols with superscripts and subscripts are used. As far as I can see, the problem is the following:
The Hamiltonian is:
H = ω₀S_z
The state of the system is, for example:
(1/√10) |+> + (3/√10) |->, where the states |+> and |-> are the spin 'up' and 'down' states in the S_z basis.
The questions that can be posed are:
1. What is the probability that a measurement of S_z yields a positive value?
2. What is the probability that, if S_z is measured and found to be positive (negative), an immediate measurement of the energy yields a positive (negative) value? -- 4 possibilities
3. What is the probability that, if S_x is measured and found to be positive (negative), an immediate measurement of the energy yields a positive (negative) value? -- 4 possibilities
4. Same question as 2 or 3 with S_y.
Have I understood the questions right?