Basic probability for quantum mechanics

[athrun200]

Homework Statement

See photo 1

Homework Equations

All formula for probability.

The Attempt at a Solution

See photo 2

It seems traditional formula (involve summation) can't be used.
So how to obtain answer for b.) and c.)?

Also, I am not sure for part a.

 

[George Jones]

Is a sum always used to calculate expectation values?

 

[athrun200]

Is a sum always used to calculate expectation values?

I think that is average value.
To get expectation values, we need to have wave function.
However, it seems there is no relation between the needle and wave function

 

[George Jones]

Let me try again.

Is a sum always used when calculating average values.

 

[paranormal]

as well.

I would like to clarify that the basic principles of probability in quantum mechanics are quite different from those in classical mechanics. In quantum mechanics, the probability of a particle being in a certain state or location is described by the wave function, which is a complex-valued function that represents the state of the particle. The square of the wave function, known as the probability density, gives the probability of finding the particle at a particular location. This is in contrast to classical mechanics, where probabilities are described by simple equations and do not involve complex numbers.

For part a), the probability of finding a particle in a particular state is given by the square of the coefficient of that state in the wave function. In this case, the wave function is given as Ψ = 2/3|1⟩ + 1/3|2⟩, so the probability of finding the particle in state |1⟩ is (2/3)^2 = 4/9, and the probability of finding it in state |2⟩ is (1/3)^2 = 1/9.

For parts b) and c), we need to use the concept of superposition, which states that a particle can exist in multiple states simultaneously. In part b), the particle is in a superposition of states |1⟩ and |2⟩, so the probability of finding it in one of these states is given by the sum of the probabilities of each individual state. In this case, the probability of finding the particle in either state is 4/9 + 1/9 = 5/9.

In part c), the particle is in a superposition of states |1⟩, |2⟩, and |3⟩, so the probability of finding it in one of these states is given by the sum of the probabilities of each individual state. In this case, the probability of finding the particle in any of these states is 4/9 + 1/9 + 0 = 5/9.

It is important to note that probabilities in quantum mechanics are not additive in the same way as in classical mechanics. Superposition allows for the possibility of finding a particle in multiple states at the same time, which is a fundamental principle of quantum mechanics. Therefore, the calculation of probabilities in quantum mechanics requires a different approach than classical mechanics.