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)

    “The real, then, is that which, sooner or later, information and reasoning would finally result in, and which is therefore independent of the vagaries of me and you. Thus, the very origin of the conception of reality shows that this conception essentially involves the notion of a COMMUNITY, without definite limits, and capable of a definite increase of knowledge.”
    —Charles Sanders Peirce (1839–1914)

    “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)

    “With these I would be.
    And with water: the waves coming forward, without cessation,
    The waves, altered by sand-bars, beds of kelp, miscellaneous
    driftwood,
    Topped by cross-winds, tugged at by sinuous undercurrents
    The tide rustling in, sliding between the ridges of stone,
    The tongues of water, creeping in, quietly.”
    —Theodore Roethke (1908–1963)