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Its Nature and Proof |
In classical physics a particle may have a location and a speed. In quantum physics a particle instead has a probability density function. In physics that probability density function is expressed in a unique way; i.e., in terms of a wave function. A wave function is a complex number valued function over space φ(X), where X is the three dimensional coordinates of a point in space. The expected value E of some quantity, say potential energy, which is a function of space coordinates, V(X) is then given by
where φ* is the complex conjugate of φ.
Because φ*φ is a probability density function it follows that
In physics there is a special notation for the expected value of a quantity V; i.e., <V>.
The reason for expressing the probability density function in the esoteric fashion of a wave function is that the wave function is determined by a special differential equation, the Schrödinger equation; i.e.,
where t is time, i is the imaginary complex unit (−1)^{½}, h is Planck's
constant divided by 2π and H is the Hamiltonian operator for the system. The Hamiltonian operator is derived from
the Hamiltonian function for the system. The Hamiltonian function for a system is its total energy expressed in terms
of its momenta and state variables. The Hamiltonian operator is derived from the function by expressing a momentum in
terms of the partial derivative with respect to its state variable. More precisely a momentum p is replaced with
ih(∂/∂x) where x is the state variable correponding to the momentum p.
Furthermore the exponential power of a momentum
becomes the order of the partial derivative. For example, a free particle in space has only kinetic energy. That
kinetic energy can be expressed as p²/(2m), where p is the linear momentum of the particle and m is its mass.
Thus the Hamiltonian function is p²/(2m) and p=m∂x/∂t, the Hamiltonian operator is
The Schrödinger equation for a free particle is then
Let V now stand for any quantity for a quantum system. The value of V and the value of the wave function at points in
space may change with time. The total derivative of the expected value of V with respect to time is the change in
There is a special function for the quantity V and the Hamiltonian operator H of the system. It is called the commutator of V and is defined as
For example, for the free particle H=ih(∂²/∂x²)/(2m) and thus
The Ehrenfest Theorem is then
This says that the time rate of change of the expected value of V is the expected value of the time rate of change
of V plus the expected value of the commutator of V and H divided by ih.
Proof:
By definition
By the Schrödinger equation ∂φ/∂t = (1/ih)Hφ and therefore
Since H involves no imaginary terms H*=H. Thus the previous relation reduces further to
Thus
When these two expressions are substituted into the equation for d <V>/dt the result is
Consider a particle in a vertical gravitational field such that the potential energy is gz, where z is vertical distance and is a parameter giving the strength of the field. The Hamiltonian function H is then
where m is the mass of the particle and p is the momentum. The linear momentum p is given by m(dz/dt). Thus the Hamiltonian operator H is
Suppose the quantity of interest is z. Then the commutator of z and H is given by
Thus the commutator of z and H is 2(∂/∂z). The expected value of the commutator is
(To be continued.)
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