Intro to Quantum Mechanics - part I
Wave properties of matter and the birth of Quantum Mechanics
Intro to Quantum Mechanics - part I
Wave properties of matter and the birth of Quantum Mechanics
When matter behaves like waves
Wave properties of matter & Intro to Quantum Mechanics
Wave properties of matter
Electron Diffraction
Recall de Broglie's result:
every entity, which we previously thought of as a particle, also exhibits wave behavior, with wavelength:
\lambda=h/p
Example: Electron diffraction
Wave properties of matter & Intro to Quantum Mechanics
Wave properties of matter
The double-slit experiment
Wave properties of matter & Intro to Quantum Mechanics
Wave properties of matter
The double-slit experiment
Wave properties of matter & Intro to Quantum Mechanics
Wave properties of matter
The double-slit experiment
What happens if we repeat this experiment, but with electrons instead of light?
Wave properties of matter & Intro to Quantum Mechanics
Wave properties of matter
The double-slit experiment
Intro to Quantum Mechanics - part I
Wave properties of matter and the birth of Quantum Mechanics
Classical Waves
Wave properties of matter & Intro to Quantum Mechanics
Wave motion
What is a wave?
Classically, a wave is defined as a traveling disturbance that carries energy.
E.g. a wave on a string is a disturbance in the vertical position of the beads travelling along the string (via tension) carrying mechanical energy.
For a traveling harmonic wave
Describe the disturbance as a function of position at a specific instant in time.
y(x,t_0)=Y \sin(\frac{2\pi}{\lambda}x+\phi_t)
where is the wavelength (distance from peak to peak in a snapshot)
\lambda
Describe the disturbance as a function of time at a specific location.
y(x_0,t)=Y \sin(\frac{2\pi}{T}t+\phi_x)
where is the period (time of one cycle, at a specific location.)
T
Wave properties of matter & Intro to Quantum Mechanics
Wave motion
The wave equation
For a traveling harmonic wave
The Classical Wave Equation!
y(x,t)=Y \sin(\frac{2\pi x}{\lambda}- \frac{2\pi t}{T}+\phi)
= Y \sin(kx- \omega t+\phi)
where:
: wave number
: angular frequency
: amplitude
: phase constant
k
\omega
Y
\phi
Q: find the relationship between
\frac{\partial^2 y}{\partial x^2}
\frac{\partial^2 y}{\partial t^2}
and
\frac{\partial^2 y}{\partial x^2} = -k^2 y
\frac{\partial^2 y}{\partial t^2} = -w^2 y
\frac{\partial^2 y}{\partial x^2} = \frac{k^2}{\omega^2} \frac{\partial^2 y}{\partial t^2}
\frac{\partial^2 y}{\partial x^2} = \frac{1}{v^2} \frac{\partial^2 y}{\partial t^2}
Intro to Quantum Mechanics - part I
Wave properties of matter and the birth of Quantum Mechanics
Wave Packets
Wave properties of matter & Intro to Quantum Mechanics
Wave motion
Superposition of two waves
Q: find the sum of two waves with different wave numbers and angular frequencies. (assume both phase angles are 90)
Y \cos(k_1x-\omega_1 t)
Y \cos(k_2 x-\omega_2 t)
+
Wave 1
Wave 2
\cos(\frac{ k_1-k_2}{2}x-\frac{ \omega_1-\omega_2}{2} t)
\cos(\frac{k_1+ k_2}{2}x-\frac{\omega_1+\omega_2}{2} t)
2Y
\times
\times
Wave properties of matter & Intro to Quantum Mechanics
Wave motion
Superposition of two waves
Y \cos(k_1x-\omega_1 t)
Y \cos(k_2 x-\omega_2 t)
+
\cos(\frac{\Delta k}{2}x-\frac{\Delta \omega}{2} t)
\cos(\frac{k_1+ k_2}{2}x-\frac{\omega_1+\omega_2}{2} t)
2Y
\times
\times
Wave 1
Wave 2
carrier
envelope
Attention: This snapshot is at t=0
Wave properties of matter & Intro to Quantum Mechanics
Wave motion
The wave packet
\psi(x,t) = \Sigma_i\ Y_i \cos(k_ix-\omega_i t)
Wave properties of matter & Intro to Quantum Mechanics
Wave motion
The wave packet
\psi(x,t) = \Sigma_i\ Y_i \cos(k_ix-\omega_i t)
Wave properties of matter & Intro to Quantum Mechanics
Wave motion
The wave packet
\psi(x,t) = \Sigma_i\ Y_i \cos(k_ix-\omega_i t)
Wave properties of matter & Intro to Quantum Mechanics
Wave motion
The wave packet
\psi(x,t) = \Sigma_i\ Y_i \cos(k_ix-\omega_i t)
Wave properties of matter & Intro to Quantum Mechanics
Wave motion
The gaussian wave packet
\psi(x,0) = A e^{-\Delta k^2\ x^2} \cos(k_0 x)
Wave properties of matter & Intro to Quantum Mechanics
Wave motion
Phase velocity and Group velocity
In a wave packet,
the velocity of some component
Y_i \cos(k_i x-\omega_i t)
is given by
v_i=\frac{\lambda_i}{T_i}= \frac{2\pi\lambda_i}{2\pi T_i}=\frac{\omega_i}{k_i}
This is known as phase velocity, because it is the rate of advance of a point with a fixed phase (e.g. crest of the wave.)
As for the full packet, the relevant velocity is that of the envelope. This is known as group velocity, and is given by
v_g=\frac{d\omega}{dk}
Wave properties of matter & Intro to Quantum Mechanics
Wave motion
Phase velocity and Group velocity
component
Y_i \cos(k_i x-\omega_i t)
v_i=\frac{\omega_i}{k_i}
phase velocity
group velocity
v_g=\frac{d\omega}{dk}
Wave properties of matter & Intro to Quantum Mechanics
Wave motion
Phase velocity and Group velocity
component
Y_i \cos(k_i x-\omega_i t)
v_i=\frac{\omega_i}{k_i}
phase velocity
group velocity
v_g=\frac{d\omega}{dk}
Intro to Quantum Mechanics - part I
Wave properties of matter and the birth of Quantum Mechanics
Wave-Particle Duality
Wave properties of matter & Intro to Quantum Mechanics
Wave-particle duality
A useful metaphor
Full moon
Crescent
The difference between these two phenomena is the setup of the observation (in this case, the position of the observer relative to the Sun-Moon direction.)
Wave properties of matter & Intro to Quantum Mechanics
Wave-particle duality
Representing particles as wave packets
\psi(x,t) = \Sigma_i\ Y_i \cos(k_ix-\omega_i t)
Wave properties of matter & Intro to Quantum Mechanics
Wave-particle duality
Group velocity & Particle velocity
Note that if the wave packet is describing a particle, then the group velocity is equivalent to
v_g = \frac{d\omega}{d k}
=\frac{d(\hbar\omega)}{d(\hbar k)}
=\frac{d(hf)}{d(h / \lambda)}
=\frac{dE}{dp}
For a free non-relativistic particle
E=\frac{1}{2}mu^2
&
p=mu
i.e.
E=\frac{p^2}{2 m}
v_g= \frac{d(p^2/2m)}{dp}=\frac{2p}{2 m}=u
then
Implying that the group velocity gives the correct particle velocity
Wave properties of matter & Intro to Quantum Mechanics
Wave properties of matter
The uncertainty principle
Wave properties of matter & Intro to Quantum Mechanics
Wave-particle duality
localization of wave packets
Recall
\cos(\frac{\Delta k}{2}x-\frac{\Delta \omega}{2} t)
\cos(\frac{k_1+ k_2}{2}x-\frac{\omega_1+\omega_2}{2} t)
2Y
\times
\times
carrier frequency
envelope
To localize the particle at any point in time, say within two adjacent points where the envelope is 0
\frac{\Delta k}{2}x_2-\frac{\Delta k}{2}x_1 = \pi
In other words
\Delta k \Delta x = 2\pi
Or, in terms of momentum
\Delta p \Delta x = 2\pi \hbar
Wave properties of matter & Intro to Quantum Mechanics
Wave properties of matter
The uncertainty principle
Wave properties of matter & Intro to Quantum Mechanics
Wave-particle duality
Heisenberg's uncertainty relations
In general,
\Delta p\ \Delta x \ge \frac{\hbar}{2}
Heisenberg's uncertainty principle.
\Delta E\ \Delta t \ge \frac{\hbar}{2}
&
Example: free particle with definite momentum
\Delta p = 0
\implies \Delta x \ge \frac{\hbar}{2 \Delta p} \approx \infty
Example: wave packet with components of varying momenta
Wave properties of matter & Intro to Quantum Mechanics
Wave properties of matter
The uncertainty principle
Wave properties of matter & Intro to Quantum Mechanics
Wave motion
Born's rule
Suppose that some entity is described by a wave packet
as shown in the figure:
What is the position of this entity?
\psi(x,t_0)
This question needs re-phrashing.
P(x>0)=\int_0^\infty|\psi(x)|^2 dx
In general, Bron's rule states
P(x_1 \rightarrow x_2) =\int_{x_1}^{x_2} |\psi(x)|^2 dx
P(-\infty \rightarrow \infty) =1
&
What is the probability of finding the entity in the range ?
x_1< x < x_2
e.g. What is the probability that the particle is at some position
x > 0
Copy of Introduction to Quantum Mechanics I
By drmoussaphysics
