Momentum operator

In quantum mechanics, the momentum operator is the operator associated with the linear momentum. The momentum operator is, in the position representation, an example of a differential operator. For the case of one particle in one dimension, the definition is:

{\hat {p}}=-i\hbar {\frac {\partial }{\partial x}}

where $ħ$ is Planck's reduced constant, $i$ the imaginary unit, and partial derivatives (denoted by $\partial$ ) are used instead of a total derivative ( $d / dx$ ) since the wave function is also a function of time. The "hat" indicates an operator. The "application" of the operator on a differentiable wave function is as follows:

{\hat {p}}\psi =-i\hbar {\frac {\partial \psi }{\partial x}}

In a basis of Hilbert space consisting of momentum eigenstates expressed in the momentum representation, the action of the operator is simply multiplication by $p$ , i.e. it is a multiplication operator, just as the position operator is a multiplication operator in the position representation. Note that the definition above is the canonical momentum, which is not gauge invariant and not a measurable physical quantity for charged particles in an electromagnetic field. In that case, the canonical momentum is not equal to the kinetic momentum.

At the time quantum mechanics was developed in the 1920s, the momentum operator was found by many theoretical physicists, including Niels Bohr, Arnold Sommerfeld, Erwin Schrödinger, and Eugene Wigner. Its existence and form is sometimes taken as one of the foundational postulates of quantum mechanics.

Origin from De Broglie plane waves

The momentum and energy operators can be constructed in the following way.^[1]

One dimension

Starting in one dimension, using the plane wave solution to Schrödinger's equation of a single free particle,

\psi (x,t)=e^{{\frac {i}{\hbar }}(px-Et)},

where $p$ is interpreted as momentum in the $x$ -direction and $E$ is the particle energy. The first order partial derivative with respect to space is

{\frac {\partial \psi (x,t)}{\partial x}}={\frac {ip}{\hbar }}e^{{\frac {i}{\hbar }}(px-Et)}={\frac {ip}{\hbar }}\psi .

This suggests the operator equivalence

{\hat {p}}=-i\hbar {\frac {\partial }{\partial x}}

so the momentum of the particle and the value that is measured when a particle is in a plane wave state is the eigenvalue of the above operator.

Since the partial derivative is a linear operator, the momentum operator is also linear, and because any wave function can be expressed as a superposition of other states, when this momentum operator acts on the entire superimposed wave, it yields the momentum eigenvalues for each plane wave component. These new components then superimpose to form the new state, in general not a multiple of the old wave function.

Three dimensions

The derivation in three dimensions is the same, except the gradient operator del is used instead of one partial derivative. In three dimensions, the plane wave solution to Schrödinger's equation is:

\psi =e^{{\frac {i}{\hbar }}(\mathbf {p} \cdot \mathbf {r} -Et)}

and the gradient is

{\begin{aligned}\nabla \psi &=\mathbf {e} _{x}{\frac {\partial \psi }{\partial x}}+\mathbf {e} _{y}{\frac {\partial \psi }{\partial y}}+\mathbf {e} _{z}{\frac {\partial \psi }{\partial z}}\\&={\frac {i}{\hbar }}\left(p_{x}\mathbf {e} _{x}+p_{y}\mathbf {e} _{y}+p_{z}\mathbf {e} _{z}\right)\psi \\&={\frac {i}{\hbar }}\mathbf {\hat {p}} \psi \end{aligned}}

where $e x, e y$ and $e z$ are the unit vectors for the three spatial dimensions, hence

\mathbf {\hat {p}} =-i\hbar \nabla

This momentum operator is in position space because the partial derivatives were taken with respect to the spatial variables.

Definition (position space)

For a single particle with no electric charge and no spin, the momentum operator can be written in the position basis as:^[2]

\mathbf {\hat {p}} =-i\hbar \nabla

where $\nabla$ is the gradient operator, $ħ$ is the reduced Planck constant, and $i$ is the imaginary unit.

In one spatial dimension this becomes:

{\hat {p}}={\hat {p}}_{x}=-i\hbar {\partial  \over \partial x}.

This is the expression for the canonical momentum. For a charged particle $q$ in an electromagnetic field, during a Gauge Transformation, the position space wave function undergoes a local^{[disambiguation needed]} U(1) group transformation^[3], and ${\hat {p}}\psi =-i\hbar {\frac {\partial \psi }{\partial x}}$ will change its value. Therefore, the canonical momentum is not gauge invariant, and hence not a measurable physical quantity.

The kinetic momentum, a gauge invariant physical quantity, can be expressed in terms of the canonical momentum, the scalar potential $φ$ and vector potential $A$ :^[4]

\mathbf {\hat {P}} =-i\hbar \nabla -q\mathbf {A}

The expression above is called minimal coupling. For electrically neutral particles, the canonical momentum is equal to the kinetic momentum.

Properties

Hermiticity

The momentum operator is always a Hermitian operator (more technically, in math terminology a "self-adjoint operator") when it acts on physical (in particular, normalizable) quantum states.^[5]

(In certain artificial situations, such as the quantum states on the semi-infinite interval [0,∞), there is no way to make the momentum operator Hermitian.^[6] This is closely related to the fact that a semi-infinite interval cannot have translational symmetry—more specifically, it does not have unitary translation operators. See below.)

Canonical commutation relation

One can easily show that by appropriately using the momentum basis and the position basis:

\left[{\hat {x}},{\hat {p}}\right]={\hat {x}}{\hat {p}}-{\hat {p}}{\hat {x}}=i\hbar .

The Heisenberg uncertainty principle defines limits on how accurately the momentum and position of a single observable system can be known at once. In quantum mechanics, position and momentum are conjugate variables.

Fourier transform

One can show that the Fourier transform of the momentum in quantum mechanics is the position operator. The Fourier transform turns the momentum-basis into the position-basis. The following discussion uses the bra–ket notation:

Let $\psi =\psi (x)$ be a wave packet $\langle \psi |\psi \rangle$ = 1, $\psi '$ the Fourier transform of $\psi$ :

\langle \psi |{\hat {p}}|\psi \rangle =h\langle \psi '|{\hat {x}}|\psi '\rangle

So momentum = h x spatial frequency, which is similar to energy = h x temporal frequency.

\langle x|{\hat {p}}|\psi \rangle =-i\hbar {\frac {d}{dx}}\psi (x)

The same applies for the position operator in the momentum basis:

\langle p|{\hat {x}}|\psi \rangle =i\hbar {\frac {d}{dp}}\psi (p)

and other useful relations:

\langle p|{\hat {x}}|p'\rangle =i\hbar {\frac {d}{dp}}\delta (p-p')

\langle x|{\hat {p}}|x'\rangle =-i\hbar {\frac {d}{dx}}\delta (x-x')

where $δ$ stands for Dirac's delta function.

Derivation from infinitesimal translations

The translation operator is denoted $T (ε)$ , where $ε$ represents the length of the translation. It satisfies the following identity:

T(\varepsilon )|\psi \rangle =\int dxT(\varepsilon )|x\rangle \langle x|\psi \rangle

that becomes

\int dx|x+\varepsilon \rangle \langle x|\psi \rangle =\int dx|x\rangle \langle x-\varepsilon |\psi \rangle =\int dx|x\rangle \psi (x-\varepsilon )

Assuming the function $ψ$ to be analytic (i.e. differentiable in some domain of the complex plane), one may expand in a Taylor series about $x$ :

\psi (x-\varepsilon )=\psi (x)-\varepsilon {\frac {d\psi }{dx}}

so for infinitesimal values of $ε$ :

T(\varepsilon )=1-\varepsilon {d \over dx}=1-{i \over \hbar }\varepsilon \left(-i\hbar {d \over dx}\right)

As it is known from classical mechanics, the momentum is the generator of translation, so the relation between translation and momentum operators is:

T(\varepsilon )=1-{i \over \hbar }\varepsilon {\hat {p}}

thus

{\hat {p}}=-i\hbar {d \over dx}.

4-momentum operator

Inserting the 3d momentum operator above and the energy operator into the 4-momentum (as a 1-form with $(+ - - -)$ metric signature):

P_{\mu }=\left({\frac {E}{c}},-\mathbf {p} \right)

obtains the 4-momentum operator;

{\hat {P}}_{\mu }=\left({\frac {1}{c}}{\hat {E}},-\mathbf {\hat {p}} \right)=i\hbar \left({\frac {1}{c}}{\frac {\partial }{\partial t}},\nabla \right)=i\hbar \partial _{\mu }

where $\partial μ$ is the 4-gradient, and the $- iħ$ becomes $+ iħ$ preceding the 3-momentum operator. This operator occurs in relativistic quantum field theory, such as the Dirac equation and other relativistic wave equations, since energy and momentum combine into the 4-momentum vector above, momentum and energy operators correspond to space and time derivatives, and they need to be first order partial derivatives for Lorentz covariance.

The Dirac operator and Dirac slash of the 4-momentum is given by contracting with the gamma matrices:

\gamma ^{\mu }{\hat {P}}_{\mu }=i\hbar \gamma ^{\mu }\partial _{\mu }={\hat {P}}=i\hbar \partial \!\!\!/

If the signature was $(- + + +)$ , the operator would be

{\hat {P}}_{\mu }=\left(-{\frac {1}{c}}{\hat {E}},\mathbf {\hat {p}} \right)=-i\hbar \left({\frac {1}{c}}{\frac {\partial }{\partial t}},\nabla \right)=-i\hbar \partial _{\mu }

instead.

References

^ Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles (2nd Edition), R. Resnick, R. Eisberg, John Wiley & Sons, 1985, ISBN 978-0-471-87373-0
^ Quantum Mechanics Demystified, D. McMahon, Mc Graw Hill (USA), 2006, ISBN 0-07-145546-9
^ Zinn-Justin, Jean; Guida, Riccardo (2008-12-04). "Gauge invariance". Scholarpedia. 3 (12): 8287. doi:10.4249/scholarpedia.8287. ISSN 1941-6016.
^ Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles (2nd Edition), R. Resnick, R. Eisberg, John Wiley & Sons, 1985, ISBN 978-0-471-87373-0
^ See Lecture notes 1 by Robert Littlejohn for a specific mathematical discussion and proof for the case of a single, uncharged, spin-zero particle. See Lecture notes 4 by Robert Littlejohn for the general case.
^ Bonneau,G., Faraut, J., Valent, G. (2001). "Self-adjoint extensions of operators and the teaching of quantum mechanics". American Journal of Physics. 69 (3): 322–331. arXiv:quant-ph/0103153. Bibcode:2001AmJPh..69..322B. doi:10.1119/1.1328351.{{cite journal}}: CS1 maint: multiple names: authors list (link)

[1] Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles (2nd Edition), R. Resnick, R. Eisberg, John Wiley & Sons, 1985, ISBN 978-0-471-87373-0

[2] Quantum Mechanics Demystified, D. McMahon, Mc Graw Hill (USA), 2006, ISBN 0-07-145546-9

[3] Zinn-Justin, Jean; Guida, Riccardo (2008-12-04). "Gauge invariance". Scholarpedia. 3 (12): 8287. doi:10.4249/scholarpedia.8287. ISSN 1941-6016.

[4] Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles (2nd Edition), R. Resnick, R. Eisberg, John Wiley & Sons, 1985, ISBN 978-0-471-87373-0

[5] See Lecture notes 1 by Robert Littlejohn for a specific mathematical discussion and proof for the case of a single, uncharged, spin-zero particle. See Lecture notes 4 by Robert Littlejohn for the general case.

[6] Bonneau,G., Faraut, J., Valent, G. (2001). "Self-adjoint extensions of operators and the teaching of quantum mechanics". American Journal of Physics. 69 (3): 322–331. arXiv:quant-ph/0103153. Bibcode:2001AmJPh..69..322B. doi:10.1119/1.1328351.{{cite journal}}: CS1 maint: multiple names: authors list (link)

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