# Basic probability for quantum mechanics

• athrun200
To calculate expectation values, a wave function is needed, but there doesn't seem to be a relationship between the needle and wave function.In summary, the conversation discusses the use of traditional formulas for probability and the difficulty in applying them to certain problems. The question also arises about the use of sums in calculating expectation values and the necessity of a wave function. However, it is uncertain how the needle and wave function are related in this scenario.
athrun200

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.

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Is a sum always used to calculate expectation values?

George Jones said:
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

Let me try again.

Is a sum always used when calculating average values.

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.

## 1. What is probability in the context of quantum mechanics?

Probability in quantum mechanics refers to the likelihood of a particular outcome occurring in a quantum system. It is represented by a complex number called the probability amplitude, which is related to the wave function of the system.

## 2. How is probability calculated in quantum mechanics?

In quantum mechanics, the probability of a particular outcome is calculated by squaring the absolute value of the probability amplitude. This is known as the Born rule and is a fundamental principle in quantum mechanics.

## 3. What is the difference between classical and quantum probability?

Classical probability is based on the principle of causality, where the outcome of a system can be determined by its initial conditions. In contrast, quantum probability is non-deterministic and relies on the superposition principle, where a system can exist in multiple states simultaneously.

## 4. How does the uncertainty principle relate to probability in quantum mechanics?

The uncertainty principle states that the more precisely we know the position of a particle, the less certain we are about its momentum, and vice versa. This is related to probability in quantum mechanics because the probability of a particle being in a particular position is related to the wave function, which also contains information about its momentum.

## 5. Can probability be used to predict the exact outcome of a quantum system?

No, probability in quantum mechanics can only provide the likelihood of a particular outcome. The actual outcome of a quantum system is determined by a random process, and thus cannot be predicted with certainty.

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