Measurement problem quantum mechanics

In summary, the conversation discusses the measurement of an electron's spin state, specifically the probability of measuring the spin up state when the electron is in a specific spin state. The conversation provides hints on how to solve the problem, including normalizing the state and using the Born rule to find the probability. The conversation also mentions the need to find the component of the spin state and how that relates to finding the probability.
  • #1
Homework Statement
Measurement

Suppose an electron is in a spin state that can be described by

|ϕ⟩=3/√2|+⟩+1/2|−⟩
where + and – are eigenstates of Sz with eigenvalue +ℏ/2 and −ℏ/2.

If we measure z-component of spin of this electron, what is the probability of measuring spin up, +ℏ/2?

Answers within 5% error will be considered correct.
Relevant Equations
I myself want to know which formula we need to use.
I was not able to attempt since I don't know which formula or method can be used to solve the problem
 
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  • #2
Ashish Somwanshi said:
Homework Statement:: Measurement

Suppose an electron is in a spin state that can be described by

|ϕ⟩=3/√2|+⟩+1/2|−⟩
where + and – are eigenstates of Sz with eigenvalue +ℏ/2 and −ℏ/2.

If we measure z-component of spin of this electron, what is the probability of measuring spin up, +ℏ/2?

Answers within 5% error will be considered correct.
Relevant Equations:: I myself want to know which formula we need to use.

I was not able to attempt since I don't know which formula or method can be used to solve the problem
First you need to normalize the state.

Hint 1: What does the result of operating ##S_z## mean?

Hint 2: Once you have that, how can you find the probability of finding the electron in the ##\mid + >## state?

-Dan
 
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  • #3
Ashish Somwanshi said:
Homework Statement:: Measurement

Suppose an electron is in a spin state that can be described by

|ϕ⟩=3/√2|+⟩+1/2|−⟩
Is that supposed to be ##\frac {\sqrt 3}{2}\ket + + \frac 1 2 \ket -##?

(Otherwise there is a rogue square root in the denominator!)
 
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  • #4
topsquark said:
First you need to normalize the state.

Hint 1: What does the result of operating ##S_z## mean?

Hint 2: Once you have that, how can you find the probability of finding the electron in the ##\mid + >## state?

-Dan
A quick note. I was looking at this wrong.

Correction:
Hint 1: How do we find the component of |+> from the ket?

Hint 2: How do we then find the probability it's in the |+> state from that?

-Dan
 
  • #5
Born rule: the probability of measuring the system ##\ket{\psi}## in state ##\ket{\alpha}## is given by
$$
\mathcal{P}(\alpha) = \left| \braket{\alpha | \psi} \right|^2
$$
 
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What is the measurement problem in quantum mechanics?

The measurement problem in quantum mechanics refers to the apparent paradox between the deterministic nature of quantum mechanics and the probabilistic outcomes observed in measurements. This raises questions about the fundamental nature of reality and the role of the observer in the measurement process.

How does the measurement problem relate to the wave-particle duality of quantum particles?

The wave-particle duality of quantum particles is a fundamental principle in quantum mechanics that states that particles can exhibit both wave-like and particle-like behavior. The measurement problem arises when attempting to determine which behavior a particle will exhibit when measured, as it seems to be influenced by the observer's actions.

What is the role of decoherence in the measurement problem?

Decoherence is the process by which a quantum system becomes entangled with its environment, leading to the collapse of its quantum state into a classical state. This is often seen as a solution to the measurement problem, as it explains how the probabilistic outcomes of measurements arise from the deterministic laws of quantum mechanics.

Are there any proposed solutions to the measurement problem?

There are various proposed solutions to the measurement problem, including the many-worlds interpretation, the Copenhagen interpretation, and the objective collapse theories. However, there is currently no consensus on which solution is the most accurate or complete.

How does the measurement problem impact our understanding of the universe?

The measurement problem highlights the limitations of our current understanding of the universe and the need for further research and development in quantum mechanics. It also raises philosophical questions about the nature of reality and the role of consciousness in the universe.

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