Introduction to Quantum Mechanics I

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

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

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)

link to notebook

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)

link to notebook

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)

link to notebook

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}

link to notebook

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}

link to notebook

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

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

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

Eg: Particle in a box

Suppose a particle with mass m is trapped inside an infinitley deep box of width L

State the boundary conditions for the allowed wave functions.
Determine the allowed (harmonic) wave functions.
Determine the energies of the allowed (harmonic) wave functions.
For the nth (harmonic) wave function, what is the most likely position to find the particle?

Since the particle is forbidden to be outside the box, the continuity of the wave function implies that it has to vanish at the boundary (box-walls):

\psi(x=0)=0

\psi(x=L)=0

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

Eg: Particle in a box

The harmonic wave functions can be expressed as

\psi(x)=A \sin\left(\frac{2\pi x}{\lambda}\right)

Satisfying the boundary conditions

\psi_n(x)=A \sin\left(\frac{n\pi x}{L}\right)

\psi (0)=\psi (L)=0 \implies L=\frac{n\lambda}{2}

Therefore

Suppose a particle with mass m is trapped inside an infinitley deep box of width L

State the boundary conditions for the allowed wave functions.
Determine the allowed (harmonic) wave functions.
Determine the energies of the allowed (harmonic) wave functions.
For the nth (harmonic) wave function, what is the most likely position to find the particle?

How do we determine A?

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

Eg: Particle in a box

\psi_n(x)=A \sin\left(\frac{n\pi x}{L}\right)

Suppose a particle with mass m is trapped inside an infinitley deep box of width L

State the boundary conditions for the allowed wave functions.
Determine the allowed (harmonic) wave functions.
Determine the energies of the allowed (harmonic) wave functions.
For the nth (harmonic) wave function, what is the most likely position to find the particle?

The normalization condition

\int_{-\infty}^{\infty} |\psi_n(x)|^2 dx=1

\int_{-\infty}^{0} +\int_{0}^{L} |\psi_n(x)|^2 dx+\int_{L}^{\infty}=1

i.e.

\int_{0}^{L} A^2 \sin^2\left(\frac{n\pi x}{L}\right)dx=1

\psi_n(x)=\sqrt{\frac{2}{L}} \sin\left(\frac{n\pi x}{L}\right)

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

Eg: Particle in a box

Suppose a particle with mass m is trapped inside an infinitley deep box of width L

State the boundary conditions for the allowed wave functions.
Determine the allowed (harmonic) wave functions.
Determine the energies of the allowed (harmonic) wave functions.
For the nth (harmonic) wave function, what is the most likely position to find the particle?

Recall that satisfying the boundary conditions required:

\lambda_n =\frac{2}{n}L

Therefore, the momentum of the particle in these harmonic states is (via de Broglie's)

p_n=h/\lambda_n=\frac{nh}{2L}

And the energies are

E_n=\tfrac{1}{2}mu^2

=\frac{p^2}{2m}

E_n=n^2\frac{h^2}{8mL^2}

n=1,2,3,...

Note that the ground state has nonzero energy! (why not?)

Also note that the ground state energy is larger for smaller boxes!

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

Eg: Particle in a box

Suppose a particle with mass m is trapped inside an infinitley deep box of width L

State the boundary conditions for the allowed wave functions.
Determine the allowed (harmonic) wave functions.
Determine the energies of the allowed (harmonic) wave functions.
For the nth (harmonic) wave function, what is the most likely position to find the particle?

\psi_n(x)=\sqrt{\frac{2}{L}} \sin\left(\frac{n\pi x}{L}\right)

|\psi_n(x)|^2=\frac{2}{L} \sin^2\left(\frac{n\pi x}{L}\right)

Recall that the harmonic wave functions have the form

Therefore, the probability density as function of position is given by

\left(\frac{n\pi x}{L}\right)=\frac{\pi}{2}+m\pi

Which is maximized at

n=1,2,3,...

m=0,1,2,..,n-1

&