Momentum Operator - Origin From de Broglie Plane Waves

Origin From De Broglie Plane Waves

The momentum and energy operators can be constructed in the following way.

One dimension

Starting in one dimension, using the plane wave solution to Schrödinger's equation:

The first order partial derivative with respect to space is

By expressing of k from the De Broglie relation:

the formula for the derivative of ψ becomes:

This suggests the operator equivalence:

so the momentum value p is a scalar factor, the momentum of the particle and the value that is measured, is the eigenvalue of the operator.

Since the partial derivative is a linear operator, the momentum operator is also linear, and because any wavefunction 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, the momenta add to the total momentum of the superimposed wave.

Three dimensions

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

and the gradient is

\begin{align} \nabla \Psi & = \bold{e}_x\frac{\partial \Psi}{\partial x} + \bold{e}_y\frac{\partial \Psi}{\partial y} + \bold{e}_z\frac{\partial \Psi}{\partial z} \\ & = i k_x\Psi\bold{e}_x + i k_y\Psi\bold{e}_y+ i k_z\Psi\bold{e}_z \\ & = \frac{i}{\hbar} \left ( p_x\bold{e}_x + p_y\bold{e}_y+ p_z\bold{e}_z \right)\Psi \\ & = \frac{i}{\hbar} \bold{\hat{p}}\Psi \\ \end{align} \,\!

where, and are the unit vectors for the three spatial dimensions, hence

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

Read more about this topic:  Momentum Operator

Famous quotes containing the words origin from, origin, plane and/or waves:

    “Our theism is the purification of the human mind. Man can paint, or make, or think nothing but man. He believes that the great material elements had their origin from his thought.”
    —Ralph Waldo Emerson (1803–1882)

    “The essence of morality is a questioning about morality; and the decisive move of human life is to use ceaselessly all light to look for the origin of the opposition between good and evil.”
    —Georges Bataille (1897–1962)

    “with the plane nowhere and her body taking by the throat
    The undying cry of the void falling living beginning to be something
    That no one has ever been and lived through screaming without enough air”
    —James Dickey (b. 1923)

    “The seashore is a sort of neutral ground, a most advantageous point from which to contemplate this world. It is even a trivial place. The waves forever rolling to the land are too far-traveled and untamable to be familiar. Creeping along the endless beach amid the sun-squall and the foam, it occurs to us that we, too, are the product of sea-slime.”
    —Henry David Thoreau (1817–1862)