5.1 X-Ray Scattering
5.2 De Broglie Waves
5.3 Electron Scattering
5.4 Wave Motion
5.5 Waves or Particles?
5.6 Uncertainty Principle
5.7 Probability, Wave Functions, and the Copenhagen Interpretation
5.8 Particle in a Box
CHAPTER 5
Wave Properties of Matter and Quantum Mechanics I
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
Many experimental observations of particles can only be understood by assigning them “wave-like” properties.
5.6: Uncertainty Principle
Remember that the localized the wave in a region ∆x requires a range of wave vectors ∆k, such that ∆x ∆k = 2π (= 1/2 for Gaussian wave packet).
It says that for any wave it is impossible to measure simultaneously, with no uncertainty, the precise values of k and x for the same particle. The wave number k may be rewritten as
This is no different for de Broglie waves: from the relation between the wave vector k and momentum p of a particle:
For the case of a Gaussian wave packet we have
In general for any wave packet,
Heisenberg’s uncertainty principle
2 2
/ k p
h p
1 2
k x p x
p x2
p x 2
HUP :
“It is impossible to know the exact position and exact momentum of an object simultaneously”.
Exact product depends on the shape of the wave packet and how ∆x and ∆p are defined (Gaussian wave packet an example of
minimum uncertainty).
Result is a natural consequence of the wave description of matter and not due to any experimental limitations.
(Example) An electron moves with a velocity of 3.6 × 106 m⋅s-1. If the accuracy of the velocity measurement is 1%, how accurately can we locate the position?
p x 2
( ~ 10 a0, not that small)
HUP for particle confinement:
Uncertainty relation has implications for confined particles.
Localizing the particle to a region, for example an electron
somewhere inside an atom, means the particle can’t have zero momentum or kinetic energy (reasonable to assume minimum momentum and energy is at least as large as uncertainty)
“minimum energy cannot be zero in an atom”
Letting ∆x ≤ d/2 implies:
p x 2
2
2md2
HUP :
Example 5.10, 5.11
Atomic nucleus has a radius of about 5×10-15m. Assume electrons are present inside the nucleus and occasionally escape,
according to HUP what range of kinetic energy is expected?
This corresponds to an energy of:
In fact, much larger than this energy observed in beta decay
It implies that the electrons do not exist in nucleus,
they must be created in the decay process.
p x 2
Energy-time Uncertainty
In addition to a relationship between ∆k and ∆x for wave localization we also showed that there is a similar relationship between ∆ω and
∆t, ∆ω∆t = 1/2 (for a Gaussian wave packet).
Given E = hf the uncertainty in angular frequency leads to an uncertainty in energy:
The energy uncertainty of a Gaussian wave packet is
In general for any wave packet, HUP for energy and time
1
E t t 2
E t 2
E t 2
HUP :
Example
Electrons in “excited states” will relax to the ground state by emitting a photon of energy E = hf i.e. Bohr atom.
If the average time the electron spends in an excited state is 1.0×10-8s, find the uncertainty in the frequency of the emitted photon.
From the HUP:
p x 2
E t 2
Line-width (∆f) measurement in experiment is most common way to estimate the lifetime in an excited state.
5.7: Probability, Wave Functions,
and the Copenhagen Interpretation
In EM waves the amplitude of the wave is the magnitude of the electric field, E and the intensity is proportional to E2.
Interference pattern in the double-slit experiment is a map of the intensity of the wave (E2).
In the case of light we saw that by decreasing the intensity of the light rather than observing the interference pattern we observed localized flashes.
in this case interference pattern predicted where more/less likely to observe a flash i.e. probability of observing photons.
For de Broglie particles how do we represent probability of finding the particle at a particular location in the wave picture?
For de Broglie particles how do we represent probability of finding the particle at a particular location in the wave picture?
In de Broglie description the particle is represented by a wave packet, Ψ(x,y,z,t).
localized using a superposition of sinusoidal waves
wave packet Ψ(x,y,z,t) is called the wave function.
In a manner similar to light at low intensity, Ψ is related to the probability of finding the particle.
IΨI2 is the probability density and represents the probability of finding the particle at a given location at a given time.
In general Ψ is complex thus:
(Definition ensures IΨI2 is positive and real)
Probability and Wave Function
The wave function determines the likelihood (or probability) of finding a particle at a particular position in 1-dimensional
space between y and y + dy at a given time.
The total probability of finding the electron must be unity.
Forcing this condition on the wave function is called normalization.
Normalization condition
The Copenhagen Interpretation
Today there is little disagreement about the mathematical formalism of quantum mechanics, such as the Schrodinger approach (Chapter 6).
Most widely accepted is the “Copenhagen school”, based on following principles:
1)
The uncertainty principle of Heisenberg 2) The complementarity principle of Bohr
3) The statistical interpretation of Born, based on probabilities determined by the wave function
Together these three concepts form a logical interpretation of the physical meaning of quantum theory. Copenhagen
interpretation
According to the Copenhagen interpretation, physics depends on
the outcomes of measurement.
The Copenhagen Interpretation
For example, in double slit experiment we can measure where the electron hits by noting the flash
the Copenhagen Interpretation rejects discussion about where the electron may have been at any time before the observation.
the measurement process randomly chooses one of the many possible outcomes allowed by the wave function and instantaneously changes to represent that outcome.
Only “reality” in physics are measurement outcomes.
Many physicists had and still do have problems with this interpretation (Einstein, Planck, de Broglie, Schrodinger):
nondeterministic nature?
instantaneous collapse of the wave function?
5.8: Particle in a Box
A particle of mass m is trapped in a one-dimensional box of width l.
The particle is treated as a wave.
The box puts boundary conditions on the wave. The wave function must be zero at the walls of the box and on the outside.
In order for the probability to vanish at the walls, we must have an integral number of half wavelengths in the box.
The energy of the particle is .
The possible wavelengths are quantized which yields the energy:
The possible energies of the particle in a box are quantized.
Particle in a Box
Results can be generalized to any confined particle.
(1) Particle cannot have arbitrary E
Allowed energies depends on nature of confinement.
(2) Confined particle cannot have zero kinetic energy.
K→0 implies p→0, and λ→∞, but confinement restricts λ ≤ 2L.
(3) Since h extremely small, quantization only apparent for extremely small m or L.
(Example) Electron in 0.1 nm box:
(Example) 10 gram marble in 10 cm box:
Amounts to a ground state velocity of:
Probability of the Particle
The probability of observing the particle between x and x + dx in each state is
Note that E0 = 0 is not a possible energy level.
For first energy level most
probable location is middle of the box (differs from classical
expectation of equal probability at all locations).
The concept of energy levels, as first discussed in the Bohr model, has surfaced in a natural way by using waves.
