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.

Theoretical Significance and History

Quantum mechanics is concerned with discrete quanta. Angular momentum eigenstates of atoms are discrete. Raising and lowering operators serve as relations between these discrete states. Ladder operators have some practical uses as shorthand methods to generate possible states from given states. They have much deeper implications for theory because they can be used to show that discreteness of angular momentum exists even in classical physics. They can be used to show the superiority of formal quantum mechanics over old quantum theory.

Many sources credit Dirac with the invention of ladder operators. Dirac's use of the ladder operators goes as far as to show that the total angular momentum quantum number $l$, or actually $j$, needs to be a non-negative half integer multiple of ħ. He also showed how the magnetic quantum number $m$ needs to run from $-j$ to $+j$ in integer steps of ħ. However, this is too general because $m$ is related to a rotation about the z-axis. It is inconceivable that a full rotation does not return a system to itself (which is the reason the introduction of spin-½ was resisted at first). In order for a full rotation to return a system to itself, $m$ must be an integer multiple of ħ, not just half-integer. In order to account for this, we split the total angular momentum of the electron into the orbital component $l$, which must obey the intuitive requirements of full rotation, and the intrinsic spin component $s$, which is not required to do so. Once this split is made, we can easily apply physical intuition to the problem. The results, then, are in agreement with experiment.

The restriction on $l$ and $m_{l}$ to integer multiples of ħ was done by "H. E. Rorschach at the 1962 Southwestern Meeting of the American Physical Society." There is also resistance to such a split, so that, in the same year, Merzbacher derived the same boundary conditions from another angle of attack, the Aharonov-Bohm effect. The arguments and ladder operators themselves have been extended many times since, to deal with spin, and to generate more than just m for given l, but also to generate l.

Terminology

Main article: Creation and annihilation operators

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 creation/annihilation operators requires the use of an annihilation operator to remove a particle from the initial state and a creation operator to add a particle to the final state.

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:

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

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

Main article: Angular momentum operator

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_-:

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

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

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:

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

Compare this result with:

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

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,

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

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

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

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

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

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

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

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

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,

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

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,

${\begin{aligned}J_{-1}^{(1)}&={\dfrac {1}{\sqrt {2}}}(J_{x}-iJ_{y})={\dfrac {J_{-}}{\sqrt {2}}}\\J_{0}^{(1)}&=J_{z}\\J_{+1}^{(1)}&=-{\dfrac {1}{\sqrt {2}}}(J_{x}+iJ_{y})=-{\dfrac {J_{+}}{\sqrt {2}}}\end{aligned}}$

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

$\mathbf {I} ^{(1)}\cdot \mathbf {J} ^{(1)}=\sum _{n=-1}^{+1}(-1)^{n}I_{n}^{(1)}J_{-n}^{(1)}=I_{0}^{(1)}J_{0}^{(1)}-I_{-1}^{(1)}J_{+1}^{(1)}-I_{+1}^{(1)}J_{-1}^{(1)},$

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

Main article: Quantum harmonic oscillator

Another application of the ladder operator concept is found in the Quantum Mechanical treatment of the Harmonic Oscillator. We can define the raising and lowering operators as

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

See also

• Creation and Annihilation Operators
• Quantum Harmonic Oscillator