Sequences of measurements in quantum mechanics

In summary, the conversation discusses calculating probabilities for energy measurements in a system with a Hamiltonian of ω0Sz and a given wavefunction. The questions include finding the probability of positive or negative results for a specific energy measurement, as well as the probability of obtaining a positive or negative energy value based on a previous measurement of Sz, Sx, or Sy. Standard symbols with superscripts and subscripts are suggested for clarity.
  • #1
ellenb899
18
0
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
 
Last edited:
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  • #2
What energy measurement are you considering here?
 
  • #3
probability of the wavefunction i have given
 
  • #4
ellenb899 said:
probability of the wavefunction i have given
That doesn't define an energy measurement.
 
  • #5
It is arbitrary. The question just asks for the probability of finding a negative or positive result for an energy measurement of the observable.
 
  • #6
ellenb899 said:
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?
 
  • #7
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
 
  • #8
What is the energy operator (the Hamiltonian) for the system? Is there a magnetic field? Which direction?
 
  • #9
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
 
  • #10
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 = ω0Sz
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 Sz basis.
The questions that can be posed are:
1. What is the probability that a measurement of Sz yields a positive value?
2. What is the probability that, if Sz 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 Sx 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 Sy.
Have I understood the questions right?
 

FAQ: Sequences of measurements in quantum mechanics

What is a sequence of measurements in quantum mechanics?

A sequence of measurements in quantum mechanics refers to the process of performing multiple measurements on a quantum system, typically one after another. Each measurement can affect the state of the system, and the outcomes of subsequent measurements depend on the results of the previous ones. This concept is crucial for understanding the dynamics and behavior of quantum systems over time.

How does the order of measurements affect the outcomes in quantum mechanics?

In quantum mechanics, the order of measurements can significantly affect the outcomes due to the non-commutative nature of quantum operators. If two observables do not commute, measuring one observable will generally alter the state of the system in such a way that the outcome of measuring the second observable will be different than if the measurements were performed in the opposite order.

What is the role of wavefunction collapse in sequences of measurements?

Wavefunction collapse plays a critical role in sequences of measurements. Upon measuring a quantum system, the wavefunction collapses to an eigenstate corresponding to the measurement outcome. This collapse changes the state of the system, which in turn affects the results of any subsequent measurements. The sequence of collapses and the resulting states are central to understanding the evolution of the system under continuous observation.

Can sequences of measurements be used to gain more information about a quantum system?

Yes, sequences of measurements can be used to gain more information about a quantum system. By carefully choosing the sequence and type of measurements, one can infer properties of the system that may not be accessible through a single measurement. Techniques such as quantum tomography and sequential measurements are employed to reconstruct the state of the system or to measure incompatible observables indirectly.

What are some practical applications of sequences of measurements in quantum mechanics?

Sequences of measurements have several practical applications in quantum mechanics, including quantum computing, quantum cryptography, and quantum error correction. In quantum computing, sequences of measurements are used to read out the results of quantum computations. In quantum cryptography, they are essential for protocols like quantum key distribution. Quantum error correction relies on sequences of measurements to detect and correct errors in quantum information processing.

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