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Lecture 3
Physics 486, Spring ‘
Lecture 3
x
Wave-Particle Duality and
Schrödinger’s Equation
Wave-Particle “Duality”
z The chief message of the last two lectures is that one
cannot classify EM radiation and matter (e.g., electrons,
protons, etc.) into distinct categories of “waves” or
“particles”
z Both radiation and matter exhibit wave-like properties
in certain situations, and particle-like properties in
others.
What does this “wave-particle” duality mean?
z When should we expect to observe wave-like properties,
and when should we expect particle-like properties?
z What does it mean for matter (e.g., electrons) to have
wave-like properties?
Before answering these questions, let’s review what we
know about waves:
Classical Wave ‘Refresher’
z We all know from classical physics that a wave is a
disturbance spread in space, which is characterized by a
wave function ψ( r ,t) that characterizes the amplitude
of the disturbance at the point r and timet.
In the case of a sound wave, ψ( r ,t) represents the
change in air pressure above the average value, while
in the case of an electromagnetic wave, ψ( r ,t) is a
component of the electric field vector E.
z The wave function ψ( r ,t) is a solution to the 2 nd^ order
“wave equation”, which is given by:
2 2 2 2 2
x v t
∂ ψ ∂ ψ ∂ ∂
2 2 2 2
v t
∂ ψ ψ ∂
Wherev is the speed of the wave (v = the speed of light, c, for electromagnetic waves), and ∇^2 is the “Laplacian operator”, which in cartesian coordinates is given by
One-dimensional wave
Three-dimensional wave
2 2 2 2 (^2) x (^2) y (^2) z
∂ ∂ ∂ ∇ = + + ∂ ∂ ∂
Classical Waves (cont.)
z We are particularly interested inplane wave solutions to
the wave equations, as these waves are periodic in space
and time:
ψ ( x , t ) = A exp i kx ( − ω t )
Where A is the amplitude of the wave, k=2π/λ is called the wavenumber, λ is the wavelength of the wave, ω=2π/T is the angular frequency of the wave, and T is the period of the wave.
The intensity of any wave is given by I = |ψ| 2 , which for a plane wave, is
simply I = |A|^2. Note that, in general, the wave function ψ(x,t) is a
complex quantity (see following ‘primer’ on complex numbers at the end).
One-dimensional plane wave
Wavelength λ
Amplitude A
A
x
Re(ψ(x,t))
Lecture 3
Wave-Particle Duality
Wave-particle duality – What makes quantum objects such as photons, electrons, neutrons, etc.,special is thecombination of particle-like and wave-like character. Indeed, Richard Feynman’s view (see his book,The Character of Physical Law (1982)) was that electrons, photons, etc., behave in their own distinctive manner, one that is outside our experience because we don’t live in the tiny regime of quantum objects.
Recall that classical waves (e.g., water, sound) are divisible, in the sense that we can split classical waves into parts. The uniqueness of the wave-particle duality associated with photons and electrons is that these objects are at once distributed over a volume of space (i.e., wavelike) but also indivisible, in the sense that we can’t split these objects into parts (i.e., particle-like).
This non-classical and unique combination of features associated with electrons and photons leads us to a physical description of quantum particles that is fundamentally indeterministic in nature! We can hopefully crystallize this idea more fully with 2 examples.
Example 2: Double Slit Experiment
z Classical particles
I(y) = I 1 (y) + I 2 (y)
S 1
S 2
y
I 1 (y) S 1
S 2
y
I 2 (y) Block slit 2 (^) Block slit 1
Open both slits:
Intensities add for classical particles!
S 1
S 2
y
I(y)
z Classical waves
I(y) = |ψ 1 +ψ 2 | 2 = |ψ 1 | 2 + |ψ 2 | 2 + (ψ 1 *ψ 2 + ψ 2 *ψ 1 ) = I 1 (y) + I 2 (y) + 2(I 1 I 2 ) 1/2cosδ
S 1
S 2
y
I 1 (y)=|ψ 1 | 2
S 1
S 2
y
Block slit 2 Block slit 1
S 1
S 2
y
I(y)
Open both slits:
Amplitudes add for classical waves!
EM wave (wavelength λ)
EM wave (wavelength λ)
EM wave (wavelength λ)
I 2 (y)=|ψ 2 | 2
I = intensity; ψ = wave amplitude
Example 2: Double Slit Experiment Two Slit Interference:
z Now, what happens if we reduce the source light (or electron beam) intensity so that only one photon or (electron) goes through at a time?
Exposure time
z Answer: Given enough time, an interference pattern will gradually build up from a huge # of seemingly random “events”!
S 1
S 2 Photons or Electrons (wavelength λ = h/p)
Lecture 3
Two-Slit Experiment Revisited
Probability Amplitude ψ = ψ 1 + ψ 2
(electron/photon can go through slit 1 or 2)
z How do we interpret the two-slit
experiment for a single incident electron
or photon in terms of the Born picture?
|ψ 1 +ψ 2 |^2
S 1
S 2
Probability, P
The fact that we must write the trajectory of the electron (photon) as a superposition of going through slit 1 AND slit 2 reflects a fundamental limitation on our ability to “know” what the electron (photon) is doing!
P = Probability of detecting a photon/electron at the screen
= |ψ 1 +ψ 2 | 2 = |ψ 1 | 2 + |ψ 2 | 2 + (ψ 1 *ψ 2 + ψ 2 *ψ 1 ) Interference term!
Optical Beamsplitter Revisited
The Born description suggests that we need to describe the probability amplitude for a photon going through
a beamsplitter as ψ = ψ 1 + ψ 2.
D
D
z Now, let’s reconsider the
beamsplitter experiment in
terms of this picture:
However,if we make a measurement, we must measure the photon either at detector D1or D2. We are forced to conclude that the measurement process changes the wavefunction, i.e., that the measurement process “collapses the wavefunction.”
Optical Beamsplitter Revisited (cont.)
D
D
We have to view wave function collapse as a non-local event, occurring instantaneously even if the wavefunction components are separated by large distances! Einstein didn’t like this “Spooky action at a distance”, but it has been demonstrated (see next slide), and is now called “entanglement”.
Los Angeles
New York
Note, however, that there is no incompatibility between instantaneous wavefunction collapse and special relativity, because the observers have no control over the events, and so they can’t use the results of the measurement for “faster than light” information exchange. For more discussion of this interesting area of quantum mechanics, take Physics 419 and/or talk to Prof. Paul Kwiat
Quantum Entanglement
Entangled photon pairs are created when a laser beam passes through a non-linear crystal (e.g., barium borate). The photon pairs created in this manner are individually unpolarized, but nevertheless have orthogonal polarizations no matter how far apart the photons are.
For more information, see Prof. Kwiat’s website: http://www.physics.uiuc.edu/People/Faculty/profiles/Kwiat/
Paul Kwiat
Lecture 3
The Schrödinger Equation
z In 1926, Erwin Schrödinger proposed an equation that described the time- and space-dependence of the ψ wavefunction for matter waves (i.e., electrons, protons,...). However, the so-called Schrodinger Equation (SEQ) was NOT derived, but was rather constructed based on a few fundamental requirements: (1). The wave-like properties of matter, in the form of the deBroglie relationship, λ = h/p, MUST be represented in the wave equation. Also, the Einstein relationship for photons E=hf (which was postulated to also hold for particles, should ALSO be included.
z Evidence for wave properties of the electron in the early 1900’s motivated an obvious question: is there a wave equation, analogous to the classical wave equation, that describes the time and position dependence of the matter wavefunction ψ(x,t)?
(2). The wave equation must be consistent with the classical relationship for energy: E = 1 / 2 p 2 /m + V, in the macroscopic limit (3). The wavefunction must be describable by the superposition principle ψ = c 1 ψ 1 + c 2 ψ 2 (e.g., so one can get constructive and destructive interference of waves). This requires that the wave equation be a linear equation!
z For the time being, let’s assume the particle is “free”, i.e., not bound by any potential (V=0). Since, by requirement (1), we demand that the deBroglie wavelength is somehow embodied in our wave equation, we might also require that:
The Schrödinger Equation
2
p
E V
m
z In order to motivate a wave equation for matter waves, let’s start with requirement (2):
h h p k
= (^) where ==h/2π and k=2π/λ is the ‘wavenumber’
z So, the energy relationship in (*) can be written:
(*)
2 2 2
p k
E
m m
z Combining this result and the previous energy relationship gives:
The Schrödinger Equation
( 2 ) 2
h E hf π f ω π
z If we also require that the Einstein relation E=hf be satisfied for particles, we get:
2 2
k
m
2
k
m
z Note that this relationship between angular frequency and wavenumber, ω~k 2 , is quite different than that for classical waves. To see this, consider the classical wave equation (which describes, for example, transverse waves on a string or longitudinal sound waves moving with a speed v): (^2 )
2 2
y 1 y
x v t
z And consider the effect of this wave equation on a wave given by: y x t ( , ) = A sin( kx − ω t )
z While the right side gives:
The Schrödinger Equation
z Plugging this wave into the left side of the classical wave equation gives:
z Consequently, the classical wave equation has a relationship between angular frequency and wavenumber given by
( )
2 2
2 sin
y
k A kx t
x
Demonstrating that the classical wave equation won’t satisfy one of the chief requirements for the matter wave equation.
( )
2 2 2 2 2
sin
y
A kx t
v t v
ω^2 = ν^2 k 2 or^ ω^2 ~^ k^2
Lecture 3
The Schrödinger Equation
z This gives two different forms for the “matter” wave equation by Schrodinger:
z A time-independent Schrodinger Equation:
2 2
2 m x^2 V^ E
z The time-dependent Schrodinger Equation:
2 2 2
i V
t m x
= Most general form
Special case
Time-Independent Schrödinger Eqn.
z What does the time-independent SEQ represent?
It’s actually not so puzzling…it’s just an expression of a familiar result:
Kinetic Energy (KE) + Potential Energy (PE) = Total Energy (E)
2 2 V x x E x dx
d x m
KE term PE term Total E term
KE
term: 2
2 2
dx
d (x) 2 m
ψ − =
The kinetic energy of the particle is associated with the curvature of the wavefunction, d 2 ψ/dx^2 ψ^ ψ ψ
ψ
ψ
2 m
p 2 m
k dx
d 2 m
kcos(kx) dx
d
Consider: cos(kx),p k
22 2 2
2 2
2 2
2
− = =
=−
∝ =
= =
=
(1). Notice for general V(x), the time-indep. SEQ has the simple form:
z For V(x) > E:
Therefore, one expects 2 general cases, for a general potential V(x).
{ } ()
2 2
2 V E x m dx
d x
2
2 x dx
d x
= + [ V x E ] m = ()−
2
2
where α
The curvature of the wavefunction is ALWAYS away from the x-axis!
z For V(x) < E: (^ )^2 () 2
2 k x dx
d x
= − [ ()]
2 k^2 = mE − Vx =
where
The curvature of the wavefunction is ALWAYS toward the x-axis! (^0)
0 x
x
Constraints on the form of ψ (x)
z Now, for the special case V(x) = constant, the SEQ has the simple form: C(x ) dx
d 2
2 = ψ ψ
For positive C (V > E), the general form of the solution is:
For negative C (V < E), the general form of the solution is:
where (^) ( ) is a constant that can be positive or negative. 2m C = (^) = 2 V − E
ψ(x) = Ae ax^ + Be -ax
2
2 2 Vx x E x dx
d x m
ψ(x) = Csinkx + Dcoskx
( V E )
m = 2 − 2 =
α
( EV )
m k = 2 − 2 =
where A, B are constants
where C, D are constants
Note that “wavenumber” k is related to the de Broglie wavelength λ by: k=2π/λ
Constraints on the form of ψ (x)
Lecture 3
Constraints on the form of ψ (x)
(3). dψ/dx must be finite, continuous* and single valued.
*Note, this requirement breaks down if the potential becomes infinite, as in the infinite square well and delta function potentials!
(2). ψ(x) must be finite, continuous and single-valued. One can show that such discontinuities cause the expectation value of the kinetic energy to be infinite, which is unphysical.
Not acceptable: ψ(x)^ is not continuous at x=0.
dx
d ψ not defined.
ψ(x)
x
ψ(x)
x
Not acceptable:
dψ/dx is not continuous at x=
Constraints on the form of ψ (x)
(4). Kinetic energy of the particle As stated previously, the kinetic energy of the particle is associated with the curvature of the wavefunction. Hence, for a given potential, the more “wiggles” the wavefunction has, the higher its kinetic energy:
ψ(x)
x
Lower energy state:
ψ(x)
x
Higher energy state:
KE
term: 2
2 2
dx
d (x)
2 m
Constraints on the form of ψ (x)
(5). Inversion symmetry of the potential If the potential experienced by a particle has inversion symmetry, i.e., V(x) = -V(x), the corresponding solutions to the SEQ will have either either even or odd parity: ψ(x) = ±ψ(-x)
ψ(x)
x
Even parity: ψ(x) = +ψ(-x)
ψ(x)
x
Odd parity: ψ(x) = -ψ(-x)
Primer on Complex Numbers
z A complex number, z, can be written generally as:
z = x + iy
1/ 2 i = − 1
where x and y are real numbers, and where
and (^) i^2 =− 1
x is called the “real part” of z (or Re(z)) and y is called the
“imaginary part” of z (or Im(z)).
The “complex conjugate” of z, written as z*, is obtained by
replacing i with –i in (*) above.
(*)
z* = x - iy
