Study Guide #5 ended with the Wave-Particle duality. At this point, a fundamental change in the understanding of the universe takes place. To truly understand and deal with the quantized nature of things will require a revision of the way things are described. This new approach is called Quantum Mechanics and extends Classical Mechanics into a discrete world. Although this is a very complicated subject, all the important ideas and basic approaches can be mastered by anyone. (The mathematical notion of subtraction can be explained, without explaining the rules of number theory, to someone who can then actually subtract!)
Read Section 39.1 … carefully!
Since there was a straight-forward connection between the energy and momentum of a photon with it’s frequency or wavelength, it seems natural to connect a wave-like character to particles that have a mass. The de Broglie wavelength of a particle is defined by
p = h/l = gmv
where l is the wavelength associated with a particular particle (of mass m) having a given speed (v), and so having a magnitude of momentum p. This is unambiguous. The energy relationship for the particle, E = hf , is trickier and must be applied with care. Should the relativistic form for the energy be used? What role does the rest-energy E0 play? How is the zero of frequency defined?
Note that particles with mass do not travel at c! So, do not apply f = c/l or E = pc to particles with mass!
A common practice is to introduce the wave-vector k. It is a vector that points in the direction of the momentum, with magnitude 2p/l.
Defining ħ = h/2p = 1.05457´10-34 J-s (pronounced “h-bar”), an alternative definition of the momentum (as a vector, denoted by bold typeface) is
p = ħk.
Although k is a measure of momentum, it has dimension of an inverse length, and can be thought of as converting the position of a particle to the equivalent phase angle of a wave.
Following the analogy of the wave-vector, using the angular frequency w = 2pf and ħ, the energy is given by
E = hf = ħw.
Read Section 39.3
The wave-like character of particles with mass requires an interpretation of just what this wave really represents. Since the wave describes the state of the particle and the particle itself is NOT oscillating, one must infer that the wave nature is describing the distribution of the particle. That is, the oscillations are in the probability of find the particle in a particular state (defined by a given energy, position, and momentum).
Because of the probability (statistical) interpretation of the wave-like nature of particles, the uncertainty (standard-deviation) of a quantity describing the state of the particle becomes important beyond an experimental issue. This means that there is a fundamental limit to how accurately quantities can be determined.
The uncertainty principle underlies all of Quantum Mechanics. It gives the minimum value for the product of uncertainties:
Position and Momentum: Dx Dpx > ħ, Dy Dpy > ħ, Dz Dpz > ħ
(for each component)
and
Energy and Time: DE Dt > ħ.
The energy-time uncertainty relation can be tricky; DE is the uncertainty in the energy of the system in a given state but Dt is NOT an uncertainty. Rather, Dt refers to the interval of time that the system remains in the given state (energy).
Read Section 39.5
The goal of physics is to
understand the universe around us. This
means that a phenomena must be described and
eventually predicted in a consistent manner.
For example, in Classical Mechanics (CM), its position and velocity
specify a particle’s motion or the evolution of the particles state. So, it’s
future is predicted by
The math required is fairly straight-forward but must be followed carefully. Keep in mind that the mathematical description contains the physical concepts.
For the classical wave picture, what is being described is the position of each part of the string at any time. For the quantum wave picture, what is being described is the wavefunction (which describes the state the particle is in) which has a position and a time component
F(r, t) = A×f(r)×j(t)
where A is a constant, the position part is described by f(r), and the time part by j(t).
Both f(r) and j(t) are, in general, oscillating functions given by
f(r) = exp(-ik×r),
j(t) = exp(-iwt) = exp(-iEt/ħ),
where the energy given by E = ħw.
Recall that a complex exponential is a sum of sines and cosines and so, the above represents generic oscillating functions.
The probability density is given by the square of the total wavefunction as
|F(r, t)|2 = |f(r)|2 |j(t)|2 = |f(r)|2
NOTE: probabilities (or probability densities) are positive-definite numbers and the sum of all probabilities must add up to one (this normalization condition determines the coefficient A).
If |F(r, t)|2 = |f(r)|2 , then the wavefunction is not an explicit function of time. In this case, F(r) represents quantum standing waves = probability density that is independent of time! (also called a stationary state).
Differential equations underlay much of the math used in physics (recall the definitions of acceleration, velocity, and position and how they show up in
F = mdp/dt = ma). Schrödinger argued (but could not prove!) that an appropriate differential equation for Quantum Mechanics has the form
¶2y(x)/¶x2 = [2m(U(x)
– E)/h2] y(x),
in one-dimension (only x-component) where y(x) is an arbitrary wavefunction. In three-dimensions, the wavefunction is dependent on the three-
components of position and Schrödinger’s Equation is given by
¶2y/¶x2 + ¶2y/¶y2 + ¶2y/¶z2 = Ñ2y(r) = [2m(U(r)
– E)/h2] y (r),
where U(r) is the potential energy (in general, also dependent on the position) and E is the total energy (thus U – E = -KE, the kinetic energy).
Note: there is no way to prove this equation. It is a hypothesis that is dimensionally correct, written in the form of a wave-equation, and has passed every
experimental test since it was introduced in 1926. One of its greatest achievement is its use in quantitatively describing the energy levels of atoms.
The Simple Harmonic Oscillator (SHO) and a particle in a box are classic examples of the application of Schrödinger’s Equation to understand the origins
of discrete energy levels.
Last Modified: 20 February 2006, GSI.