# Ladder operator

A ladder operator is an operator in quantum mechanics that maps an eigenstate of another operator, typically a Hamiltonian or an angular momentum component, to another eigenstate of the same operator whose eigenvalue differs by a fixed step. The pair consisting of a raising (creation) operator and a lowering (annihilation) operator lets the spectrum and the states be found by algebra alone, without solving differential equations.<sup>[1](https://www.maths.dur.ac.uk/users/kasper.peeters/mathphys/harmonic_oscillator.html)</sup><sup> • </sup><sup>[2](https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Quantum_Mechanics_%28Fowler%29/03%253A_Mostly_1-D_Quantum_Mechanics/3.04%253A_The_Simple_Harmonic_Oscillator)</sup>

| Key fact | Detail |
|---|---|
| Oscillator definition | \( a = (m\omega\hat{x} + i\hat{p})/\sqrt{2\hbar m\omega} \), \( a^{\dagger} = (m\omega\hat{x} - i\hat{p})/\sqrt{2\hbar m\omega} \); neither is Hermitian, and \( a^{\dagger} \) is the adjoint of \( a \)<sup>[1](https://www.maths.dur.ac.uk/users/kasper.peeters/mathphys/harmonic_oscillator.html)</sup> |
| Commutation relation | \( [a, a^{\dagger}] = 1 \), the Heisenberg algebra, also called the standard bosonic commutation relation<sup>[1](https://www.maths.dur.ac.uk/users/kasper.peeters/mathphys/harmonic_oscillator.html)</sup><sup> • </sup><sup>[3](https://engineering.purdue.edu/CQEM/teaching/Roth_fqt_lecture_notes_fall_2022.pdf)</sup> |
| Hamiltonian form | \( \hat{H} = \hbar\omega(N + \tfrac{1}{2}) \) with the number operator \( N = a^{\dagger}a \)<sup>[2](https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Quantum_Mechanics_%28Fowler%29/03%253A_Mostly_1-D_Quantum_Mechanics/3.04%253A_The_Simple_Harmonic_Oscillator)</sup> |
| Spectrum | \( E_n = \hbar\omega(n + \tfrac{1}{2}) \) for \( n = 0, 1, 2, \ldots \); the ladder is semi-infinite, with a ground state but no upper limit<sup>[1](https://www.maths.dur.ac.uk/users/kasper.peeters/mathphys/harmonic_oscillator.html)</sup><sup> • </sup><sup>[4](https://physics.umd.edu/courses/Phys402/AnlageSpring09/Lecture%205%20Summary.pdf)</sup> |
| Action on states | \( a^{\dagger}\phi_n = \sqrt{n+1}\,\phi_{n+1} \) and \( a\,\phi_n = \sqrt{n}\,\phi_{n-1} \), so \( \phi_n = (a^{\dagger})^n\phi_0/\sqrt{n!} \)<sup>[1](https://www.maths.dur.ac.uk/users/kasper.peeters/mathphys/harmonic_oscillator.html)</sup> |
| Angular momentum form | \( L_{\pm} = L_x \pm iL_y \), with \( [L_z, L_{\pm}] = \pm\hbar L_{\pm} \) and \( [L_{\pm}, \mathbf{L}^2] = 0 \)<sup>[5](https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Advanced_Quantum_Mechanics_%28Kok%29/07%3A_Orbital_Angular_Momentum_and_Spin/7.01%3A_Orbital_Angular_Momentum)</sup> |
| Angular momentum action | \( L_{\pm}\lvert l, m\rangle = \hbar\sqrt{l(l+1) - m(m\pm 1)}\,\lvert l, m\pm 1\rangle \)<sup>[5](https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Advanced_Quantum_Mechanics_%28Kok%29/07%3A_Orbital_Angular_Momentum_and_Spin/7.01%3A_Orbital_Angular_Momentum)</sup> |

## The harmonic oscillator construction

For a one-dimensional oscillator of mass \( m \) and angular frequency \( \omega \), the Hamiltonian is \( \hat{H} = \hat{p}^2/2m + \tfrac{1}{2}m\omega^2\hat{x}^2 \). After scaling position and momentum to dimensionless variables \( \xi = x\sqrt{m\omega/\hbar} \) and \( \pi = p/\sqrt{\hbar m\omega} \), the Hamiltonian becomes \( H = \frac{\hbar\omega}{2}(\pi^2 + \xi^2) \). [Paul Dirac](https://www.edgepedia.org/paul-dirac)'s idea was to factorize this sum of squares: the naive product \( (\xi + i\pi)(\xi - i\pi) \) fails because the operators do not commute, but the symmetrized version works, and it leads to the definitions of \( a \) and \( a^{\dagger} \) above. The LibreTexts account notes that this factorization of a second-order differential operator into first-order operators is close to the idea behind the [Dirac equation](https://www.edgepedia.org/dirac-equation).<sup>[2](https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Quantum_Mechanics_%28Fowler%29/03%253A_Mostly_1-D_Quantum_Mechanics/3.04%253A_The_Simple_Harmonic_Oscillator)</sup><sup> • </sup><sup>[6](https://site.physics.georgetown.edu/~jkf/grad_quant2/html/lecture3.html)</sup>

**The algebra.** From the canonical commutator \( [x, p] = i\hbar \) it follows that \( [a, a^{\dagger}] = 1 \), while \( [a, a] = [a^{\dagger}, a^{\dagger}] = 0 \); this is the [Heisenberg algebra](https://www.edgepedia.org/heisenberg-algebra). The Hamiltonian then takes the two equivalent forms \( \hat{H} = \hbar\omega(a^{\dagger}a + \tfrac{1}{2}) = \hbar\omega(aa^{\dagger} - \tfrac{1}{2}) \). The commutators with the Hamiltonian are \( [\hat{H}, a] = -\hbar\omega\, a \) and \( [\hat{H}, a^{\dagger}] = \hbar\omega\, a^{\dagger} \), and these determine the spectrum.<sup>[1](https://www.maths.dur.ac.uk/users/kasper.peeters/mathphys/harmonic_oscillator.html)</sup>

**Why the spectrum follows.** If \( \phi \) is an eigenfunction with energy \( E \), then \( \hat{H}\,a^{\dagger}\phi = (E + \hbar\omega)\,a^{\dagger}\phi \) and \( \hat{H}\,a\phi = (E - \hbar\omega)\,a\phi \). By induction, \( (a^{\dagger})^n\phi \) has energy \( E + n\hbar\omega \) and \( a^n\phi \) has energy \( E - n\hbar\omega \). For this reason \( a^{\dagger} \) and \( a \) are called creation and annihilation operators: they create and annihilate energy in units of \( \hbar\omega \).<sup>[1](https://www.maths.dur.ac.uk/users/kasper.peeters/mathphys/harmonic_oscillator.html)</sup>

The downward climb cannot continue indefinitely, because normalizable eigenfunctions must have positive energy. There must therefore be a state \( \phi_0 \) annihilated by \( a \), satisfying \( a\phi_0 = 0 \). Since \( N = a^{\dagger}a \) has only non-negative eigenvalues, because \( \langle\psi|N|\psi\rangle = \langle\psi_a|\psi_a\rangle \ge 0 \), the lowest state has eigenvalue \( n = 0 \), and the ladder values are the non-negative integers. The ground-state energy is \( \tfrac{1}{2}\hbar\omega \), so even the lowest state carries finite energy, and the full spectrum is \( E_n = \hbar\omega(n + \tfrac{1}{2}) \).<sup>[1](https://www.maths.dur.ac.uk/users/kasper.peeters/mathphys/harmonic_oscillator.html)</sup><sup> • </sup><sup>[2](https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Quantum_Mechanics_%28Fowler%29/03%253A_Mostly_1-D_Quantum_Mechanics/3.04%253A_The_Simple_Harmonic_Oscillator)</sup>

**Normalization and wavefunctions.** Requiring each \( \phi_n \) to be normalized fixes the step factors \( C_n = \sqrt{n} \), giving \( \phi_n = (a^{\dagger})^n\phi_0/\sqrt{n!} \) and the actions \( a^{\dagger}\phi_n = \sqrt{n+1}\,\phi_{n+1} \), \( a\phi_n = \sqrt{n}\,\phi_{n-1} \). The ground state follows from the first-order differential equation \( (m\omega x + \hbar\,\partial_x)\phi_0 = 0 \), whose normalized solution is the Gaussian \( \phi_0(x) = (m\omega/\pi\hbar)^{1/4} e^{-m\omega x^2/2\hbar} \); this state saturates Heisenberg's uncertainty principle.

**Practical use.** Because \( \hat{x} = \sqrt{\hbar/2m\omega}(a + a^{\dagger}) \) and \( \hat{p} = -i\sqrt{\hbar m\omega/2}(a - a^{\dagger}) \), expectation values of polynomials in \( x \) and \( p \) can be computed from the known ladder actions and orthonormality, without integrals. For example \( \langle x \rangle = 0 \) and \( \langle x^2 \rangle = (\hbar/m\omega)(n + \tfrac{1}{2}) \) in the \( n \)-th state.<sup>[1](https://www.maths.dur.ac.uk/users/kasper.peeters/mathphys/harmonic_oscillator.html)</sup>

## Angular momentum ladders

The same construction applies to angular momentum. Because the components do not commute, one works with simultaneous eigenstates \( |l, m\rangle \) of \( \mathbf{L}^2 \) and \( L_z \), and defines \( L_{\pm} = L_x \pm iL_y \) with \( L_- = L_{+}^{\dagger} \). The commutation relations are \( [L_z, L_{\pm}] = \pm\hbar L_{\pm} \), \( [L_+, L_-] = 2\hbar L_z \), and \( [L_{\pm}, \mathbf{L}^2] = 0 \), so \( L_{\pm} \) raise or lower the \( m \) eigenvalue by one unit of \( \hbar \) while leaving \( l \) unchanged.<sup>[5](https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Advanced_Quantum_Mechanics_%28Kok%29/07%3A_Orbital_Angular_Momentum_and_Spin/7.01%3A_Orbital_Angular_Momentum)</sup><sup> • </sup><sup>[7](https://www.physics.rutgers.edu/~steves/Mark_and_Janos/Lectures/Lecture_17_Angular_Momentum.pdf)</sup>

**A finite ladder.** Unlike the oscillator, the angular momentum ladder ends in both directions: \( L_+|l, l\rangle = 0 \) and \( L_-|l, -l\rangle = 0 \), because an infinite increase of \( |m| \) would make the expectation value of \( L_x^2 + L_y^2 \), a non-negative observable, negative. Counting the states from \( -l \) to \( +l \) in steps of one gives \( 2l + 1 \) states, so \( l \) must be an integer or half-integer, and \( \mathbf{L}^2 \) has eigenvalue \( \hbar^2 l(l+1) \). The matrix elements follow from the algebra: \( L_{\pm}|l, m\rangle = \hbar\sqrt{l(l+1) - m(m\pm 1)}\,|l, m\pm 1\rangle \).<sup>[5](https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Advanced_Quantum_Mechanics_%28Kok%29/07%3A_Orbital_Angular_Momentum_and_Spin/7.01%3A_Orbital_Angular_Momentum)</sup><sup> • </sup><sup>[8](https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Essential_Graduate_Physics_-_Quantum_Mechanics_%28Likharev%29/05%3A_Some_Exactly_Solvable_Problems/5.06%3A_Revisiting_Spherically-symmetric_Problems)</sup>

The parallel with the oscillator is close enough that the angular momentum ladder operators are described as acting exactly like creation and annihilation operators, moving the system up or down a ladder of eigenstates. [Julian Schwinger](https://www.edgepedia.org/julian-schwinger) built an explicit model of angular momentum out of two independent oscillators, connecting the \( a_{\pm} \), \( a_{\pm}^{\dagger} \) operators of uncoupled oscillators to the \( J \) algebra.<sup>[8](https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Essential_Graduate_Physics_-_Quantum_Mechanics_%28Likharev%29/05%3A_Some_Exactly_Solvable_Problems/5.06%3A_Revisiting_Spherically-symmetric_Problems)</sup><sup> • </sup><sup>[9](https://www.physics.umd.edu/courses/Phys622/agashe/F20/notes/lecture34.pdf)</sup>

## Wider role

The commutation relation \( [a, a^{\dagger}] = 1 \) is called the standard bosonic commutation relation, after the bosons, particles with integer spin, whose creation and annihilation operators obey it.

The method generalizes. A general theorem states that if two operators \( A \) and \( B \) have a commutator \( [A, B] = c \) that is a number, then \( e^{\lambda B} \) acting on an eigenstate of \( A \) produces another eigenstate with eigenvalue shifted by \( \lambda c \); for the oscillator this reproduces the ladder of energies separated by \( \hbar\omega \), while for other choices it can generate a continuum of eigenstates. Annihilation-creation operators have also been defined in a unified way for most solvable single-degree-of-freedom quantum systems, including discrete quantum mechanics, as the positive and negative frequency parts of the exact Heisenberg operator solution.<sup>[10](https://galileo.phys.virginia.edu/classes/751.mf1i.fall02/CoherentStates.htm)</sup><sup> • </sup><sup>[11](https://arxiv.org/abs/quant-ph/0605215)</sup>

## References

1. [Simple Harmonic Oscillator, lecture notes by Kasper Peeters, Durham University](https://www.maths.dur.ac.uk/users/kasper.peeters/mathphys/harmonic_oscillator.html)
2. [3.4: The Simple Harmonic Oscillator, Physics LibreTexts (Fowler)](https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Quantum_Mechanics_%28Fowler%29/03%253A_Mostly_1-D_Quantum_Mechanics/3.04%253A_The_Simple_Harmonic_Oscillator)
3. [Fundamentals of Quantum Technology, lecture notes, Purdue University](https://engineering.purdue.edu/CQEM/teaching/Roth_fqt_lecture_notes_fall_2022.pdf)
4. [Lecture 5 Highlights, Phys 402, University of Maryland](https://physics.umd.edu/courses/Phys402/AnlageSpring09/Lecture%205%20Summary.pdf)
5. [7.1: Orbital Angular Momentum, Physics LibreTexts (Kok)](https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Advanced_Quantum_Mechanics_%28Kok%29/07%3A_Orbital_Angular_Momentum_and_Spin/7.01%3A_Orbital_Angular_Momentum)
6. [Phys 506 lecture 3, Georgetown University](https://site.physics.georgetown.edu/~jkf/grad_quant2/html/lecture3.html)
7. [Intermediate Quantum Mechanics, Lecture 17: Angular Momentum, Rutgers University](https://www.physics.rutgers.edu/~steves/Mark_and_Janos/Lectures/Lecture_17_Angular_Momentum.pdf)
8. [5.6: Revisiting Spherically-symmetric Problems, Physics LibreTexts (Likharev)](https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Essential_Graduate_Physics_-_Quantum_Mechanics_%28Likharev%29/05%3A_Some_Exactly_Solvable_Problems/5.06%3A_Revisiting_Spherically-symmetric_Problems)
9. [Schwinger's oscillator model for angular momentum, Phys 622 notes, University of Maryland](https://www.physics.umd.edu/courses/Phys622/agashe/F20/notes/lecture34.pdf)
10. [Coherent States, Physics 751 notes, University of Virginia](https://galileo.phys.virginia.edu/classes/751.mf1i.fall02/CoherentStates.htm)
11. [Unified Theory of Annihilation-Creation Operators for Solvable ('Discrete') Quantum Mechanics, arXiv:quant-ph/0605215](https://arxiv.org/abs/quant-ph/0605215)

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*Topic: Encyclopedia › Physical world and mathematics › Physics › Quantum physics › Quantum mechanics › Quantum formalism and states › Operators, observables, and angular momentum › Ladder operators and algebraic solution methods*

*Initially written Sep 17, 2026 · Reviewed: Sep 17, 2026; Sep 30, 2026; Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

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