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:

    “All good poetry is the spontaneous overflow of powerful feelings: it takes its origin from emotion recollected in tranquillity.”
    —William Wordsworth (1770–1850)

    “There are certain books in the world which every searcher for truth must know: the Bible, the Critique of Pure Reason, the Origin of Species, and Karl Marx’s Capital.”
    —W.E.B. (William Edward Burghardt)

    “It was the most ungrateful and unjust act ever perpetrated by a republic upon a class of citizens who had worked and sacrificed and suffered as did the women of this nation in the struggle of the Civil War only to be rewarded at its close by such unspeakable degradation as to be reduced to the plane of subjects to enfranchised slaves.”
    —Anna Howard Shaw (1847–1919)

    “During our twenties...we act toward the new adulthood the way sociologists tell us new waves of immigrants acted on becoming Americans: we adopt the host culture’s values in an exaggerated and rigid fashion until we can rethink them and make them our own. Our idea of what adults are and what we’re supposed to be is composed of outdated childhood concepts brought forward.”
    —Roger Gould (20th century)