Ladder operator

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.

Terminology

There is some confusion regarding the relationship between the raising and lowering ladder operators and the creation and annihilation operators commonly used in quantum field theory. The creation operator a_i^† increments the number of particles in state i, while the corresponding annihilation operator a_i decrements the number of particles in state i. This clearly satisfies the requirements of the above definition of a ladder operator: the incrementing or decrementing of the eigenvalue of another operator (in this case the particle number operator).

Confusion arises because the term ladder operator is typically used to describe an operator that acts to increment or decrement a quantum number describing the state of a system. To change the state of a particle with the creation/annihilation operators of QFT requires the use of both an annihilation operator to remove a particle from the initial state and a creation operator to add a particle to the final state.

The term "ladder operator" is also sometimes used in mathematics, in the context of the theory of Lie algebras and in particular the affine Lie algebras, to describe the su(2) subalgebras, from which the root system and the highest weight modules can be constructed by means of the ladder operators.^[1] In particular, the highest weight is annihilated by the raising operators; the rest of the positive root space is obtained by repeatedly applying the lowering operators (one set of ladder operators per subalgebra).

General formulation

Suppose that two operators X and N have the commutation relation,

$[N,X] = cX,\quad$

for some scalar c. If $\scriptstyle{|n\rangle}$ is an eigenstate of N with eigenvalue equation,

$N|n\rangle = n|n\rangle, \,$

then the operator X acts on $\scriptstyle{|n\rangle}$ in such a way as to shift the eigenvalue by c:

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

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

If N is a Hermitian operator then c must be real and the Hermitian adjoint of X obeys the commutation relation:

$[N,X^\dagger] = -cX^\dagger.\quad$

In particular, if X is a lowering operator for N then X^† is a raising operator for N and vice versa.

Angular momentum

A particular application of the ladder operator concept is found in the quantum mechanical treatment of angular momentum. For a general angular momentum vector, J, with components, J_x, J_y and J_z we define the two ladder operators, J₊ and J_–:^[2]

$J_+ = J_x + iJ_y,\quad$

$J_- = J_x - iJ_y,\quad$

where i is the imaginary unit.

The commutation relation between the cartesian components of any angular momentum operator is given by

$[J_i,J_j] = i\hbar\epsilon_{ijk}J_k,$

where ε_ijk is the Levi-Civita symbol and each of i, j and k can take any of the values x, y and z. From this the commutation relations between the ladder operators and J_z can easily be obtained:

$\left[J_z,J_\pm\right] = \pm\hbar J_\pm.\quad$

$\left[J_+, J_-\right] = 2\hbar J_z.\quad$

The properties of the ladder operators can be determined by observing how they modify the action of the J_z operator on a given state:

$\begin{align} J_zJ_\pm|j\,m\rangle &= \left(J_\pm J_z + \left[J_z, J_\pm\right] \right) |j\,m\rangle\\ &= \left(J_\pm J_z \pm \hbar J_\pm\right)|j\,m\rangle\\ &= \hbar\left(m \pm 1\right)J_\pm|j\,m\rangle. \end{align}$

Compare this result with:

$J_z|j\,m\pm 1\rangle = \hbar(m\pm 1)|j\,m\pm 1\rangle.\quad$

Thus we conclude that $\scriptstyle{J_\pm|j\,m\rangle}$ is some scalar multiplied by $\scriptstyle{|j\,m\pm 1\rangle}$,

$J_+|j\,m\rangle = \alpha|j\,m+1\rangle,\quad$

$J_-|j\,m\rangle = \beta|j\,m-1\rangle.\quad$

This illustrates the defining feature of ladder operators in quantum mechanics: the incrementing (or decrementing) of a quantum number, thus mapping one quantum state onto another. This is the reason that they are often known as raising and lowering operators.

To obtain the values of α and β we first take the norm of each operator, recognizing that J₊ and J_- are a Hermitian conjugate pair ($\scriptstyle{J_\pm = J_\mp^\dagger}$),

$\langle j\,m|J_+^\dagger J_+|j\,m\rangle = \langle j\,m|J_-J_+|j\,m\rangle = \langle j\,m+1|\alpha^*\alpha|j\,m+1\rangle = |\alpha|^2$,

$\langle j\,m|J_-^\dagger J_-|j\,m\rangle = \langle j\,m|J_+J_-|j\,m\rangle = \langle j\,m-1|\beta^*\beta|j\,m-1\rangle = |\beta|^2$.

The product of the ladder operators can be expressed in terms of the commuting pair J² and J_z,

$J_-J_+ = (J_x - iJ_y)(J_x + iJ_y) = J_x^2 + J_y^2 + i[J_x,J_y] = J^2 - J_z^2 - \hbar J_z,$

$J_+J_- = (J_x + iJ_y)(J_x - iJ_y) = J_x^2 + J_y^2 - i[J_x,J_y] = J^2 - J_z^2 + \hbar J_z.$

Thus we can express the values of |α|² and |β|² in terms of the eigenvalues of J² and J_z,

$|\alpha|^2 = \hbar^2j(j+1) - \hbar^2m^2 - \hbar^2m = \hbar^2(j-m)(j+m+1),$

$|\beta|^2 = \hbar^2j(j+1) - \hbar^2m^2 + \hbar^2m = \hbar^2(j+m)(j-m+1).$

The phases of α and β are not physically significant, thus they can be chosen to be real and we have:^[3]

$J_+|j\,m\rangle = \hbar\sqrt{(j-m)(j+m+1)}|j\,m+1\rangle = \hbar\sqrt{j(j+1)-m(m+1)}|j\,m+1\rangle,$

$J_-|j\,m\rangle = \hbar\sqrt{(j+m)(j-m+1)}|j\,m-1\rangle = \hbar\sqrt{j(j+1)-m(m-1)}|j\,m-1\rangle.$

Confirming that m is bounded by the value of j ($\scriptstyle{-j\leq m\leq j}$) we have:

$J_+|j\,j\rangle = 0, \,$

$J_-|j\,-j\rangle = 0. \,$

The above demonstration is effectively the construction of the Clebsch-Gordan coefficients.

Applications in atomic and molecular physics

Many terms in the Hamiltonians of atomic or molecular systems involve the scalar product of angular momentum operators. An example is the magnetic dipole term in the hyperfine Hamiltonian,^[4]

$\hat{H}_\text{D} = \hat{A}\mathbf{I}\cdot\mathbf{J}, \quad$

where I is the nuclear spin. Angular momentum algebra can often be simplified by recasting it in the spherical basis. Using the notation of spherical tensor operators, the "-1", "0" and "+1" components of J⁽¹⁾ ≡ J are given by,^[5]

$\begin{align} J_{-1}^{(1)} &= \dfrac{1}{\sqrt{2}}(J_x - iJ_y) = \dfrac{J_-}{\sqrt{2}}\\ J_0^{(1)} &= J_z\\ J_{+1}^{(1)} &= -\frac{1}{\sqrt{2}}(J_x + iJ_y) = -\frac{J_+}{\sqrt{2}}. \end{align}$

From these definitions it can be shown that the above scalar product can be expanded as

$\mathbf{I}^{(1)}\cdot\mathbf{J}^{(1)} = \sum_{n=-1}^{+1}(-1)^nI_{n}^{(1)}J_{-n}^{(1)} = I_0^{(1)}J_0^{(1)} - I_{-1}^{(1)}J_{+1}^{(1)} - I_{+1}^{(1)}J_{-1}^{(1)},$

The significance of this expansion is that it clearly indicates which states are coupled by this term in the Hamiltonian, that is those with quantum numbers differing by m_i = ±1 and m_j = ∓1 only.

Harmonic oscillator

Another application of the ladder operator concept is found in the quantum mechanical treatment of the harmonic oscillator. We can define the lowering and raising operators as

$\begin{align} a &=\sqrt{m\omega \over 2\hbar} \left(\hat x + {i \over m \omega} \hat p \right) \\ a^{\dagger} &=\sqrt{m \omega \over 2\hbar} \left(\hat x - {i \over m \omega} \hat p \right) \end{align}$

They provide a convenient means to extract energy eigenvalues without directly solving the system's differential equation.

History

Many sources credit Dirac with the invention of ladder operators.^[6] Dirac's use of the ladder operators shows that the total angular momentum quantum number $j$ needs to be a non-negative half integer multiple of ħ.

See also

• Creation and annihilation operators
• Quantum harmonic oscillator

References

1. Fuchs, Jurgen (1992), Affine Lie Algebras and Quantum Groups, Cambridge University Press, ISBN 0-521-48412-X
2. de Lange, O. L.; R. E. Raab (1986). "Ladder operators for orbital angular momentum". American Journal of Physics 54 (4): 372–375. Bibcode:1986AmJPh..54..372D. doi:10.1119/1.14625.
3. Sakurai, Jun J. (1994). Modern Quantum Mechanics. Delhi, India: Pearson Education, Inc. p. 192. ISBN 81-7808-006-0.
4. Woodgate, Gordon K. (1983-10-06). Elementary Atomic Structure. ISBN 978-0-19-851156-4. Retrieved 2009-03-03.
5. "Angular Momentum Operators". Graduate Quantum Mechanics Notes. University of Virginia. Retrieved 2009-04-06.
6. http://www.fisica.net/quantica/quantum_harmonic_oscillator_lecture.pdf