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Quantum Mechanics
In quantum mechanics, the dynamics of a single particle of mass m acted on by a conservative force is determined by the Schrödinger equation
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Here
V(r) is the potential associated with the force and
is the reduced Planck’s constant. The probability density, P(r),
that the particle is located at position r is determined from the wave
function,
, by
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where
 is the complex conjugate of
.
To get some idea of the significance of the Schrödinger equation and the wave function, consider a free particle for which V(r) = 0. Now the Schrödinger equation has the form of a diffusion equation
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with
diffusion coefficient
. Hence
is a measure of the rate of spreading of the wave function. Classically a particle
at rest at a definite position r0 remains at rest at the same
position. This corresponds to the limit
. The wave function in this case is zero everywhere except at r0.
If
then the ‘uncertainty’ in position grows with time. This wave
packet spreading is illustrated below:
As
time progresses the width of P(x) increases
. This width is proportional to the standard deviation of the position x.
In quantum mechanics the standard deviation of the probability distribution
of a variable is often called the uncertainty
in the variable. The reduced Planck’s constant
has the value 1.05 10-34 J s. If we have an object of mass 1 kg
then the time scale for the uncertainty in the position of this object to increase
by about 10-6 m is estimated to be 3 106 years. Hence
classical physics is amply adequate to describe the dynamics of macroscopic
objects.
However on the atomic scale, classical physics fails. The photoelectric effect, Compton scattering, and double slit diffraction experiments show that photons have both wave-like and particle-like properties. Similarly, Thomson’s experiments on cathode rays and the electron double slit experiment show that electrons also have both wave-like and particle like properties. This led to the idea of wave-particle duality. Different experiments elicit different behavior but both a wave-based picture and a particle-based picture are necessary and complementary. The melding of these two pictures is achieved by quantum mechanics.
A
‘particle’ can be described as a wave with all the dynamic information of
the particle being contained in the wave function
. Consider a one-dimensional monochromatic plane wave
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(we
assume that all waves can be built-up from such plane waves). Here k
is the wave number and
is the angular frequency of the wave. From the photoelectric effect we know
that the energy of a photon with frequency
is
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To relate this to the wave function, we note that
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Introducing the energy, we find
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Hence the energy is associated with the operator
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Also for a wave (e.g. a light wave) the momentum is related to the wave number by
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Since
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we find
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so that the momentum is associated with the operator
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For a free particle the energy is purely kinetic and is related to the momentum of the particle by
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If we consider this equation in terms of the operators and apply it to the wave function we find
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which is the Schrödinger equation for a free particle.
If we include a potential energy in the total energy
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we have
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which is the Schrödinger equation for a particle moving in one dimension acted on by a conservative force. The solution is in general a function of position and time.
We might want to predict the momentum of a particle at a particular time. However, due to the probabilistic nature of quantum mechanics, measurement of the momentum in identical systems will give a distribution of values. What can be done is to find the expectation value of the momentum
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from the wave function. Here it is assumed that the wave function is normalized so that
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A more general form of the Schrödinger equation is
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where H is the Hamiltonian operator. In many cases (but not all) the Hamiltonian operator is the same as that for the total energy. For a single particle acted on by conservative forces and viewed in an inertial reference frame
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The boundary conditions are that the wave function is zero on the boundary of the region accessible to the particle. This region might be all space, in which case the wave function must be zero at infinity.