Quantum mechanics. Wave Properties of Matter and Quantum Mechanics I. 5.1X-Ray Scattering 5.2De Broglie Waves 5.3Electron Scattering 5.4Wave Motion 5.5Waves or Particles? 5.6Uncertainty Principle 5.7Probability, Wave Functions, and the Copenhagen Interpretation
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Wave Properties of Matter and Quantum Mechanics I
Louis de Broglie (1892-1987)
I thus arrived at the overall concept which guided my studies: for both matter and radiations, light in particular, it is necessary to introduce the corpuscle concept and the wave concept at the same time.
- Louis de Broglie, 1929
5.1: X-Ray Scattering
Max von Laue suggested that if x-rays were a form of electromagnetic radiation, interference effects should be observed.
Crystals act as three-dimensional gratings, scattering the waves and producing observable interference effects.
Bragg’s Law: nλ = 2d sin θ(n = integer)
William Lawrence Bragg interpreted the x-ray scattering as the reflection of the incident x-ray beam from a unique set of planes of atoms within the crystal.
There are two conditions for constructive interference of the scattered x rays:
The Bragg Spectrometer
A Bragg spectrometer scatters x rays from crystals. The intensity of the diffracted beam is determined as a function of scattering angle by rotating the crystal and the detector.
When a beam of x rays passes through a powdered crystal, the dots become a series of rings.
In 1925, Davisson and Germer experimentally observed that electrons were diffracted (much like x-rays) in nickel crystals.
George P. Thomson (1892–1975), son of J. J. Thomson, reported seeing electron diffraction in transmission experiments on celluloid, gold, aluminum, and platinum. A randomly oriented polycrystalline sample of SnO2 produces rings.
If a light-wave could also act like a particle, why shouldn’t matter-particles also act like waves?
Louis V. de Broglie(1892-1987)
electron de Broglie wavelength
Multiplying by p/2p, we find the angular momentum:
It will actually be different, but, in some cases, the solutions are the same.
Y(x,t) = A exp[i(kx – wt – q)]
C. Jönsson of Tübingen, Germany, succeeded in 1961 in showing double-slit interference effects for electrons by constructing very narrow slits and using relatively large distances between the slits and the observation screen.
This experiment demonstrated that precisely the same behavior occurs for both light (waves) and electrons (particles).
Each photon actually goes through both slits!
When you block one slit, the one-slit pattern returns.
At low intensities, Young’s two-slit experiment shows that light propagates as a wave and is detected as a particle.
Need lph < d to distinguish the slits.
Diffraction is significant only when the aperture is ~ the wavelength of the wave.
The momentum of the photons used to determine which slit the electron went through is enough to strongly modify the momentum of the electron itself—changing the direction of the electron! The attempt to identify which slit the electron passes through will in itself change the diffraction pattern! Electrons also propagate as waves and are detected as particles.
The act of making one measurement perturbs the other.
Precisely measuring the time disturbs the energy.
Precisely measuring the position disturbs the momentum.
The Heisenbergmobile. The problem was that when you looked at the speedometer you got lost.
The average of a positive quantity must always exceed its uncertainty:
The probability of the particle being between x1 and x2 is given by:
The total probability of finding the particle is 1. Forcing this condition on the wave function is called normalization.
Quantum Mechanics II
Erwin Schrödinger (1887-1961)
A careful analysis of the process of observation in atomic physics has shown that the subatomic particles have no meaning as isolated entities, but can only be understood as interconnections between the preparation of an experiment and the subsequent measurement.
- Erwin Schrödinger
I think it is safe to say that no one understands quantum mechanics. Do not keep saying to yourself, if you can possibly avoid it, “But how can it be like that?” because you will get “down the drain” into a blind alley from which nobody has yet escaped. Nobody knows how it can be like that.
- Richard Feynman
Those who are not shocked when they first come across quantum mechanics cannot possibly have understood it.
- Niels Bohr
Richard Feynman (1918-1988)
6.1: The Schrödinger Wave Equation
where V = V(x,t)
The Schrödinger wave equation in its time-dependent form for a particle of energy E moving in a potential V in one dimension is:
where i is the square root of -1.
The Schrodinger Equation is THE fundamental equation of Quantum Mechanics.
General Solution of the Schrödinger Wave Equation when V = 0
Try this solution:
This works as long as:
which says that the total energy is the kinetic energy.
General Solution of the Schrödinger Wave Equation when V = 0
In free space (with V = 0), the general form of the wave function is
which also describes a wave moving in the x direction. In general the amplitude may also be complex.
The wave function is also not restricted to being real. Notice that this function is complex.
Only the physically measurable quantities must be real. These include the probability, momentum and energy.
Normalization and Probability
The probability P(x) dx of a particle being between x and x + dx is given in the equation
The probability of the particle being between x1 and x2 is given by
The wave function must also be normalized so that the probability of the particle being somewhere on the x axis is 1.
The left side depends only on t, and the right side depends only on x. So each side must be equal to a constant. The time dependent side is:
We integrate both sides and find:
where C is an integration constant that we may choose to be 0.
But recall our solution for the free particle:
In which f(t) = e -iw t, so: w = B / ħ or B = ħw, which means that: B = E !
So multiplying by y(x), the spatial Schrödinger equation becomes:
This equation is known as the time-independent Schrödinger wave equation, and it is as fundamental an equation in quantum mechanics as the time-dependent Schrodinger equation.
So often physicists write simply:
is an operator.
If there are an infinite number of possibilities, and x is continuous:
And the expectation of some function of x, g(x):
The position x is its own operator. Done.
Energy operator: The time derivative of the free-particle wave function is:
Substituting w = E / ħ yields
The energy operator is:
The expectation value of the energy is:
The energy is:
½ - ½ cos(2npx/L)
The Schrödinger equation outside the finite well in regions I and III is:
Considering that the wave function must be zero at infinity, the solutions for this equation are
Consider the Taylor expansion of a potential function:
Consider the second-order term of the Taylor expansion of a potential function:
Substituting this into Schrödinger’s equation:
Let and which yields:
The wave function solutions are where Hn(x) are Hermite polynomials of order n.
As the quantum number increases, however, the solution approaches the classical result.
The zero point energy is called the Heisenberg limit:
The quantum mechanical result is one of the most remarkable features of modern physics. There is a finite probability that the particle can penetrate the barrier and even emerge on the other side!
The wave function in region II becomes:
The transmission probability that describes the phenomenon of tunneling is:
This violation of classical physics is allowed by the uncertainty principle. The particle can violate classical physics by DE for a short time, Dt ~ ħ / DE.
So the three-dimensional Schrödinger wave equation is
It’s easy to show that:
When the box is a cube:
Try 10, 4, 3 and 8, 6, 5
Note that more than one wave function can have the same energy.