# QM: Expectation value of raising and lowering operator

• barefeet
In summary, the equation ##\langle j,m_z \mid J_-J_+ \mid j,m_z \rangle = h^2 [j(j+1)-m_z^2]## is missing a term due to the fact that ##J_x## does not commute with ##J_y## when computing ##J_-J_+##.
barefeet

## Homework Statement

Using
$$J^2 \mid j,m_z \rangle = h^2 j(j+1) \mid j,m_z \rangle$$
$$J_z \mid j,m_z \rangle = hm_z \mid j,m_z \rangle$$

Derive that :
$$\langle j,m_z \mid J_-J_+ \mid j,m_z \rangle = h^2[ j(j+1) - m_z(m_z+1)]$$

## Homework Equations

$$J_- = J_x - iJ_y$$
$$J_+ = J_x + iJ_y$$

## The Attempt at a Solution

$$J_-J_+ = (J_x- iJ_y)(J_x + iJ_y) = J_x^2 + J_y^2 = J^2 - J_z^2$$

$$\langle j,m_z \mid J_-J_+ \mid j,m_z \rangle = \langle j,m_z \mid J^2 - J_z^2 \mid j,m_z \rangle = h^2[ j(j+1) - m_z^2]$$

Apparently I am missing a term here but I don't know where it should come from. I thought this should be true
$$J_z^2 \mid j,m_z \rangle = J_zJ_z \mid j,m_z \rangle = h^2m_z^2 \mid j,m_z \rangle$$
(Note: h is hbar everywhere )

You are assuming that ##J_x## commutes with ##J_y## when computing ##J_-J_+##. This is not the case.

barefeet

## What is the raising and lowering operator in quantum mechanics?

The raising and lowering operator, also known as the ladder operator, is a mathematical operator used in quantum mechanics to describe the energy states of a quantum system. It is denoted by a+ and a and is used to move up or down the energy levels of a system.

## What is the expectation value of the raising and lowering operator?

The expectation value of the raising and lowering operator is a measure of the average energy of a quantum system in a particular state. It is calculated by taking the inner product of the state vector with the operator and then squaring the result.

## How is the expectation value of the raising and lowering operator related to the energy of a system?

The expectation value of the raising and lowering operator is directly related to the energy of a system. This is because the operator is used to describe the energy states of a system, and the expectation value is a measure of the average energy of a system in a particular state.

## What is the significance of the expectation value of the raising and lowering operator in quantum mechanics?

The expectation value of the raising and lowering operator is a fundamental concept in quantum mechanics as it allows us to calculate the average energy of a system in a particular state. It also helps us understand how the energy levels of a system change when the operator is applied.

## Can the expectation value of the raising and lowering operator have a negative value?

Yes, the expectation value of the raising and lowering operator can have a negative value. This can occur when the quantum system is in a superposition of energy states, where the average energy is a combination of positive and negative values.

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