Ladder operators

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In linear algebra (and its application to quantum mechanics), a raising or lowering operator (collectively known as ladder operators) is an operator that increases or decreases the eigenvalue of another operator. In quantum mechanics, the raising operator is sometimes called the creation operator, and the lowering operator the annihilation operator. Well-known applications of ladder operators in quantum mechanics are in the formalisms of the quantum harmonic oscillator and angular momentum.

Suppose that two operators X and N have the commutation relation

$[N,X] = cX\,$

for some scalar c. Then the operator X will act in such a way as to shift the eigenvalue of an eigenstate of N by c:

$NX\|n\rangle$	${}= (XN+[N,X])\|n\rangle$
	${} = (XN + cX)\|n\rangle$
	${} = XN\|n\rangle + cX\|n\rangle$
	${} = Xn\|n\rangle + cX\|n\rangle$
	${} = (n+c)X\|n\rangle$

In other words, if $|n\rangle$ is an eigenstate of $N$ with eigenvalue n then $X|n\rangle$ is an eigenstate of N with eigenvalue n + c. A raising operator for N is an operator X for which c is real and positive and a lowering operator is one for which c is real and negative.