# Matrix Representation of the Angular Momentum Raising Operator

• I
• hnicholls
In summary, L+ with l = 1 and m = -1, 0, 1, the matrix element column 3 row 2 is incorrectly calculated to be sqrt{2}.
hnicholls
TL;DR Summary
Matrix Representation of the Angular Momentum Raising Operator. Calculating L(+).
In calculating the matrix elements for the raising operator L(+) with l = 1 and m = -1, 0, 1 each of my elements conforms to a diagonal shifted over one column with values [(2)^1/2]hbar on that diagonal, except for the element, L(+)|0,-1>, where I have a problem.

This should be value [(2)^1/2]hbar; however, I get L(+)|0,-1> = (0(0+1)-(-1)((-1)+1))^1/2|0,-1+1> = 0hbar|0,0>. This would be a 0 value not [(2)^1/2]hbar. Not sure where I making my mistake.

Please use ## ( beginning and end) for Latex rendeting to make post easier to read.

L+ with l = 1 and m = -1, 0, 1

L+|0,-1> = (0(0+1)-(-1)((-1)+1))^1/2|0,-1+1> = 0ħ|0,0>. This would be a 0 value not √2ħ. Not sure why this wrong.

L+ with l = 1 and m = -1, 0, 1

L+|0,-1> = √[0(0+1)-(-1)((-1)+1)]|0,-1+1> = 0ħ|0,0>. This would be a 0 value not √2ħ. Not sure why this wrong.

In the following I use ##\hbar=1## and write ##|m \rangle## (we are in the subspace with ##l=1## anyway). Then the "raising operator" acts on these basis states by
$$\hat{L}_+|1 \rangle=0, \quad \hat{L}_+ |0 \rangle=\sqrt{2}|1 \rangle, \quad \hat{L}_{+} |-1 \rangle=\sqrt{2} |0 \rangle.$$
Here I used the general formula
$$|\hat{L}_+ |l,m \rangle=\sqrt{(l-m)(l+m+1)} |l,m+1 \rangle.$$
The matrix representation with the basis ##|1 \rangle##, ##0 \rangle##, ##|-1 \rangle## (in that order) thus reads
$$\hat{L}_+=\begin{pmatrix} 0 & \sqrt{2} & 0 \\ 0 & 0 & \sqrt{2} \\ 0 & 0 &0 \end{pmatrix}$$

Mentz114
Applying the general formula

$$|\hat{L}_+ |l,m \rangle=\sqrt{(l-m)(l+m+1)} |l,m+1 \rangle.$$

$$|\hat{L}_+ |l,m \rangle=\sqrt{(0-(-1))(0+(-1)+1)} |l,m+1 \rangle.$$

$$|\hat{L}_+ |l,m \rangle=\sqrt{(+1)(0)} |0,0 \rangle.$$

$$|\hat{L}_+ |l,m \rangle=\sqrt{0} |0,0 \rangle.$$

I don't see how matrix element column 3 row 2 can be$$\sqrt{2}$$I still get 0. Thank you for the response.

hnicholls said:
Applying the general formula

You calculated this wrong. These states all have ##l = 1##, not ##l = 0##. Try the formula with ##l = 1##.

vanhees71
Got it! Thanks! Did not correctly apply l = 1 as the subspace and the m values (1,0,-1) as the basis states.

vanhees71

## 1. What is the matrix representation of the angular momentum raising operator?

The matrix representation of the angular momentum raising operator is a square matrix that describes the action of the operator on a quantum state. It is commonly denoted as J+ and is used to calculate the angular momentum of a particle in quantum mechanics.

## 2. How is the matrix representation of the angular momentum raising operator derived?

The matrix representation of the angular momentum raising operator is derived from the commutation relations of the angular momentum operators. These relations involve the position and momentum operators and are used to determine the matrix elements of J+.

## 3. What is the significance of the matrix representation of the angular momentum raising operator?

The matrix representation of the angular momentum raising operator is significant because it allows us to perform calculations and make predictions about the behavior of quantum systems. It is a fundamental tool in quantum mechanics and is used in a variety of applications, such as studying atomic and molecular structure.

## 4. How does the matrix representation of the angular momentum raising operator relate to the quantum mechanical spin?

The matrix representation of the angular momentum raising operator is closely related to the quantum mechanical spin of a particle. In fact, for spin-1/2 particles, the matrix representation of J+ is equivalent to the Pauli spin matrices. This allows us to use the matrix representation to study spin states and their interactions.

## 5. Can the matrix representation of the angular momentum raising operator be applied to any quantum system?

Yes, the matrix representation of the angular momentum raising operator can be applied to any quantum system, as long as the system has angular momentum. This includes particles with spin, as well as systems with orbital angular momentum. However, the specific form of the matrix representation may vary depending on the system being studied.

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