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ARoyC

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- What is a measurement in Quantum Mechanics? Is the outcome definite or probabilistic if I apply a particular measurement operator? How is it different from Classical measurements?

Hi.

This is Annwoy Roy Choudhury. I have just completed my first-year undergraduate studies in Physics. I am new to Quantum Mechanics. There are certain confusions I have regarding Quantum Measurements. It would be really kind of you to help me out.

Postulate 3 states,

1. For known |ψ〉, say if I apply a Measurement Operator Mm, then will I always get the projection of |ψ〉on |m〉? Or will it be probabilistic? Can the measurement outcome be a projection on some other basis state? What will I actually get from the measurement? An example would be beneficial.

2. What happens for unknown |ψ〉?

3. What is essentially the difference between Classical Measurements and Quantum Measurements?

Thank you for your time.

Sincere Regards

Annwoy

This is Annwoy Roy Choudhury. I have just completed my first-year undergraduate studies in Physics. I am new to Quantum Mechanics. There are certain confusions I have regarding Quantum Measurements. It would be really kind of you to help me out.

Postulate 3 states,

An example is,Quantum measurements are described by a collection {Mm} of measurement operators. These are operators acting on the state space of the system being measured. The index m refers to the measurement outcomes that may occur in the experiment. If the state of the quantum system is |ψ〉immediately before the measurement then the probability that result m occurs is given by,

p(m) = 〈ψ|Mm† Mm|ψ〉

and the state of the system after the measurement is

Mm |ψ〉/ √(〈ψ|Mm† Mm|ψ〉 )

Let's come to my questions.A measurement on a single qubit with two outcomes defined by the two measurement operators M0 = |0〉〈0| and M1 = |1〉〈1|

Then the probability of obtaining measurement outcome 0 is

p(0) = 〈ψ|M0†M0|ψ〉 = 〈ψ|M0|ψ〉 = |a|^2

Similarly, the probability of obtaining the measurement outcome 1 is p(1) = |b|^2. The state after measurement in the two cases is therefore

M0|ψ〉/ |a| = (a/|a|) |0〉

M1|ψ〉/ |b| = (b/|b|) |1〉

1. For known |ψ〉, say if I apply a Measurement Operator Mm, then will I always get the projection of |ψ〉on |m〉? Or will it be probabilistic? Can the measurement outcome be a projection on some other basis state? What will I actually get from the measurement? An example would be beneficial.

2. What happens for unknown |ψ〉?

3. What is essentially the difference between Classical Measurements and Quantum Measurements?

Thank you for your time.

Sincere Regards

Annwoy