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)

    “Each structure and institution here was so primitive that you could at once refer it to its source; but our buildings commonly suggest neither their origin nor their purpose.”
    —Henry David Thoreau (1817–1862)

    “In time the scouring of wind and rain will wear down the ranges and plane off the region until it has the drab monotony of the older deserts. In the meantime—a two-million-year meantime—travelers may enjoy the cruel beauties of a desert in its youth,....”
    —For the State of California, U.S. public relief program (1935-1943)

    “You waves, though you dance by my feet like children at play,
    Though you glow and you glance, though you purr and you dart;
    In the Junes that were warmer than these are, the waves were more gay,
    When I was a boy with never a crack in my heart.”
    —William Butler Yeats (1865–1939)