Diffraction and uncertainty
• When a photon passes through a narrow slit, its momentum becomes
uncertain and the photon can deflect to either side.
• A diffraction pattern is the result of many photons
hitting the screen.
Diffraction and uncertainty
• These images record the positions where individual photons in a two-slit
interference experiment strike the screen.
• As more photons reach the screen, a recognizable
QuickCheck
A beam of photons passes through a narrow slit. The photons land on a distant screen, forming a diffraction pattern. In order for a particular photon to land at the center of the diffraction pattern, it must pass
A. through the center of the slit.
B. through the upper half of the slit. C. through the lower half of the slit.
The Heisenberg uncertainty principle
• You cannot simultaneously know the position and
momentum of a photon, or any other particle, with arbitrarily great precision.
• The better you know the value of one quantity, the less well you know the value of the other.
• There is a similar uncertainty relationship for the y- and z-coordinate axes and their corresponding momentum
Uncertainty in energy
• There is also an uncertainty principle that involves energy and time.
• The better we know a photon’s energy, the less certain we are of when we will observe the photon:
The Schrödinger equation in 1-D
• In a one-dimensional model, a quantum-mechanical particle is described by a wave function (x, t).
• The one-dimensional Schrödinger equation for a free particle of mass m is:
• The presence of i (the square root of –1) in the Schrödinger equation means that wave functions are always complex functions.
The Schrödinger equation in 1-D: A free
particle
• A free particle can have a definite momentum p = ħk and energy
E = ħω.
• Such a particle is not localized at all: The wave function extends to infinity.
Example 1 – The wave function
is a superposition of two free-particle wave functions. Show that this wave function satisfies the Schrödinger equation for a free particle of mass m.
The Schrödinger equation in 1-D: Wave
packets
• Superposing a large number of sinusoidal waves with different wave numbers and appropriate amplitudes can produce a wave pulse that has a wavelength
λav = 2π/kav and is localized within a region of space of length Δx.
• Shown are the real and
The Schrödinger equation in 1-D: Wave
packets
• The resulting probability distribution has only one maximum.
• This localized pulse has
aspects of both particle and wave.
• It is a particle in the sense that it is localized in space; if we look from a distance, it may look like a point.
• But it also has a periodic structure that is characteristic of a wave.
The Schrödinger equation in 1-D
• If a particle of mass m moves in the presence of a potential energy function U(x), the one-dimensional Schrödinger
equation for the particle is:
The Schrödinger equation in 1-D: Stationary
states
•
If a particle has a definite energy E, the wave function
is a product of a time-independent wave function
and a factor that depends on time t but not position:
•
For such a stationary state the probability distribution
The Schrödinger equation in 1-D: Stationary
states
• The time-independent one-dimensional Schrödinger equation for a stationary state of energy E is:
Example 2 – Consider the wave function
Is this a valid time-independent wave function for a free particle in a stationary state? What is the energy
corresponding to this wave function?
Newtonian view of a particle in a box
• Let’s look at a simple model in which a particle is bound so that it cannot escape to infinity, but rather is confined to a restricted region of space.
• Our system consists of a particle confined
Potential energy for a particle in a box
• The potential energy corresponding to the rigid walls is infinite, so the particle cannot escape.
Particle in a box: Wave functions, energy
levels
• The energy levels for a particle in a box are:
Particle in a box: Wave functions, energy
levels
Particle in a box: Probability and
normalization
• Shown are the first three stationary-state wave functions for a particle in a box (a) and the associated probability distribution functions (b).
• There are locations where there is zero probability of finding the particle.
• Wave functions must be normalized so that the integral of over all x
equals 1 (which means there is 100% probability of finding the particle
Particle in a finite potential well
• A finite well is a potential well that has straight sides but finite height.
Particle in a finite potential well
• Shown are the stationary-state wave functions and corresponding energies for one particular finite well.
Particle in a finite potential well
• Shown are graphs of the
probability distributions for the first three bound states of a finite well.
• As with the infinite well, not all positions are equally
likely.
Example 3 – Show that is a solution of the time-independent Schrödinger equation outside a finite well of height Uo. What happens to the function in the limit the well is infinite?