In quantum mechanics, momentum is defined as an operator on the wave function. 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.
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For a single particle with no electric charge and no spin, the momentum operator can be written in the position basis as: [1]
where ∇ is the gradient operator, ħ is the reduced Planck constant, and i is the imaginary unit.
In one spatial dimension this becomes:
This is a commonly encountered form of the momentum operator, though not the most general one. For a charged particle q in an electromagnetic field, described by the scalar potential φ and vector potential A, the momentum operator must be replaced by the kinetic momentum operator, which includes a contribution from the A field: [2]
where is the canonical momentum operator given as the usual momentum operator:
so the momentum operator generalizes to
This is of course true for electrically charged or neutral particles, since the second term vanishes if q is zero and the original operator appears.
The momentum operator is always a Hermitian operator when it acts on physical (in particular, normalizable) quantum states.[3]
One can easily show that by appropriately using the momentum basis and the position basis:
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 same applies for the Position operator in the momentum basis:
and other useful relations:
where stands for Dirac's delta function.
The momentum and energy operators can easily be constructed in the following way. [4] Starting in one dimension, using the plane wave solution to Schrödinger's equation:
The first order partial derivative with respect to space is
by the De Broglie relation:
we have
The momentum value p is a scalar factor, the momentum of the particle and the value that is mesured. Cancellation of Ψ leads to
The partial derivative is a linear operator so this expression can only be the momentum operator:
So it can be concluded that the scalar p is the eigenvalue of the operator, while is the operator. Summarizing these results:
The derivation in three dimensions is the same, except using the gradient operator instead of one partial derivative. In three dimensions, the plane wave solution to Schrödinger's equation is:
The gradeint is
hence
Since the operators are linear, they are valid for any linear combination of plane waves, and so they can act on any wavefunction without affecting the properties of the wavefunction or operators.
Suppose we have an infinitesimal translation operator , where represents the length of the infinitesimal translation, then
that becomes
We assume the function to be analytic (or simply differentiable, for simplicity), so we may write:
so we have for infinitesimal values of epsilon:
we know from classical mechanics that momentum is the generator of translation, so we know that
thus
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