Ladder operator

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).

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

${\displaystyle [N,X]=cX,\quad }$

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

${\displaystyle N|n\rangle =n|n\rangle ,\,}$

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

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

In other words, if ${\displaystyle \scriptstyle {|n\rangle }}$ is an eigenstate of N with eigenvalue n then ${\displaystyle \scriptstyle {X|n\rangle }}$ is an eigenstate of N with eigenvalue n + c or it is zero. 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:

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

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

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]

${\displaystyle J_{+}=J_{x}+iJ_{y},\quad }$

${\displaystyle J_{-}=J_{x}-iJ_{y},\quad }$

where i is the imaginary unit.

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

${\displaystyle [J_{i},J_{j}]=i\hbar \epsilon _{ijk}J_{k},}$

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:

${\displaystyle \left[J_{z},J_{\pm }\right]=\pm \hbar J_{\pm }.\quad }$

${\displaystyle \left[J_{+},J_{-}\right]=2\hbar J_{z}.\quad }$

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:

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

Compare this result with:

${\displaystyle J_{z}|j\,(m\pm 1)\rangle =\hbar (m\pm 1)|j\,(m\pm 1)\rangle .\quad }$

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

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

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

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 (${\displaystyle \scriptstyle {J_{\pm }=J_{\mp }^{\dagger }}}$),

${\displaystyle \langle j\,m|J_{+}^{\dagger }J_{+}|j\,m\rangle =\langle j\,m|J_{-}J_{+}|j\,m\rangle =\langle j\,(m+1)|\alpha ^{*}\alpha |j\,(m+1)\rangle =|\alpha |^{2}}$,

${\displaystyle \langle j\,m|J_{-}^{\dagger }J_{-}|j\,m\rangle =\langle j\,m|J_{+}J_{-}|j\,m\rangle =\langle j\,(m-1)|\beta ^{*}\beta |j\,(m-1)\rangle =|\beta |^{2}}$.

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

${\displaystyle J_{-}J_{+}=(J_{x}-iJ_{y})(J_{x}+iJ_{y})=J_{x}^{2}+J_{y}^{2}+i[J_{x},J_{y}]=J^{2}-J_{z}^{2}-\hbar J_{z},}$

${\displaystyle J_{+}J_{-}=(J_{x}+iJ_{y})(J_{x}-iJ_{y})=J_{x}^{2}+J_{y}^{2}-i[J_{x},J_{y}]=J^{2}-J_{z}^{2}+\hbar J_{z}.}$

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

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

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

The phases of α and β are not physically significant, thus they can be chosen to be positive and real (Condon-Shortley phase convention). We then have:[3]

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

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

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

${\displaystyle J_{+}|j\,j\rangle =0,\,}$

${\displaystyle J_{-}|j\,(-j)\rangle =0.\,}$

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

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

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

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

Another application of the ladder operator concept is found in the quantum mechanical treatment of the electronic energy of hydrogen-like atoms and ions [6]. We can define the lowering and raising operators (based on the Laplace–Runge–Lenz classical vector)

${\displaystyle {\vec {A}}=\left({\frac {1}{Ze^{2}\mu }}\right)\left\{{\vec {L}}\times {\vec {p}}-{\boldsymbol {i}}\hbar {\vec {p}}\right\}+{\frac {r}{\vec {r}}}}$

where ${\displaystyle {\vec {L}}}$ is the angular momentum, ${\displaystyle {\vec {p}}}$ is the linear momentum, ${\displaystyle \mu }$ is the reduced mass of the system, ${\displaystyle e}$ is the electronic charge, and ${\displaystyle Z}$ is the atomic number of the nucleus. Analogous to the angular momentum ladder operators, one has ${\displaystyle A_{+}=A_{x}+iA_{y}}$ and ${\displaystyle A_{-}=A_{x}-iA_{y}}$.

The commutators needed to proceed are:

${\displaystyle [A_{\pm },L_{z}]=\mp {\boldsymbol {i}}\hbar A_{\mp }}$

and

${\displaystyle [A_{\pm },L^{2}]=\mp 2\hbar ^{2}A_{\pm }-2\hbar A_{\pm }L_{z}\pm 2\hbar A_{z}L_{\pm }}$.

Therefore,

${\displaystyle A_{+}|?,\ell ,m_{\ell }>\rightarrow |?,\ell ,m_{\ell }+1>}$

and

${\displaystyle -L^{2}\left(A_{+}|?,\ell ,\ell >\right)=-\hbar ^{2}(\ell +1)((\ell +1)+1)\left(A_{+}|?,\ell ,\ell >\right)}$

so

${\displaystyle A_{+}|?,\ell ,\ell >\rightarrow |?,\ell +1,\ell +1>}$

where the "?" indicates a nascent quantum number which emerges from the discussion.

Given the Pauli[7] equations Pauli Equation IV:

${\displaystyle 1-A\cdot A=-\left({\frac {2E}{\mu Z^{2}e^{4}}}\right)(L^{2}+\hbar ^{2})}$

and Pauli Equation III:

${\displaystyle \left(A\times A\right)_{j}=-\left({\frac {2{\boldsymbol {i}}\hbar E}{\mu Z^{2}e^{4}}}\right)L_{j}}$

and starting with the equation

${\displaystyle A_{-}A_{+}|\ell ^{*},\ell ^{*}>=0}$

and expanding, one obtains (assuming ${\displaystyle \ell ^{*}}$ is the maximum value of the angular momentum quantum number consonant with all other conditions),

${\displaystyle \left(1+{\frac {2E}{\mu Z^{2}e^{4}}}(L^{2}+\hbar ^{2})-i{\frac {2i\hbar E}{\mu Z^{2}e^{4}}}L_{z}\right)|?,\ell ^{*},\ell ^{*}>=0}$

which leads to the infamous (Rydberg_formula)

${\displaystyle E_{n}=-{\frac {\mu Z^{2}e^{4}}{2\hbar ^{2}(\ell ^{*}+1)^{2}}}}$

implying that ${\displaystyle \ell ^{*}+1\rightarrow n\rightarrow ?}$, where ${\displaystyle n}$ is the traditional quantum number.

Many sources credit Dirac with the invention of ladder operators.[8] Dirac's use of the ladder operators shows that the total angular momentum quantum number ${\displaystyle j}$ needs to be a non-negative half integer multiple of ħ.

• Creation and annihilation operators
• Quantum harmonic oscillator