Unit 4 - Notes

PHY109

Unit 4: Quantum Mechanics

1. Need for Quantum Mechanics

By the end of the 19th century, Classical Mechanics (Newtonian mechanics and Maxwell’s electromagnetic theory) successfully explained macroscopic phenomena (motion of planets, projectiles, electricity, optics). However, it failed to explain microscopic phenomena at the atomic and sub-atomic levels.

Major failures of Classical Mechanics that led to Quantum Mechanics:

Stability of the Atom: According to classical electrodynamics, an accelerating charged particle (electron) should radiate energy continuously and spiral into the nucleus. This contradicted the observed stability of atoms.
Spectral Lines: Classical theory predicted a continuous spectrum for atoms, but experimental observation showed discrete line spectra (e.g., Hydrogen spectrum).
Black Body Radiation: Classical theory failed to explain the energy distribution in the spectrum of a black body (Ultraviolet Catastrophe).
Photoelectric Effect: Classical wave theory could not explain why light below a certain frequency failed to eject electrons, regardless of intensity.
Variation of Specific Heat: Classical theory could not explain the drop in specific heat of solids at very low temperatures.

Conclusion: A new formulation, Quantum Mechanics, was developed based on the dual nature of matter (wave-particle duality) and the quantization of energy.

2. Black Body Radiation

Definition: A black body is an ideal body that absorbs all electromagnetic radiation incident upon it, regardless of frequency or angle of incidence. When heated, it emits radiation known as black body radiation.

Spectral Distribution Curves

Experimental observations of the energy distribution ( $E_\lambda$ ) versus wavelength ( $\lambda$ ) at different temperatures ( $T$ ) reveal:

Energy is not distributed uniformly; it varies with wavelength.
At a given temperature, energy increases with $\lambda$ , reaches a maximum at $\lambda_{max}$ , and then decreases.
Wien’s Displacement Law: The wavelength corresponding to maximum energy ( $\lambda_{max}$ ) is inversely proportional to absolute temperature ( $T$ ).
$\lambda_{max} T = \text{constant}$
Stefan-Boltzmann Law: The total energy emitted is proportional to the fourth power of temperature ( $E \propto T^4$ ).

The Failure of Classical Theory (Rayleigh-Jeans Law)

Rayleigh and Jeans attempted to explain this using classical thermodynamics and standing waves.

Prediction: Energy density $E_\lambda \propto 1/\lambda^4$ .
Result: This worked for long wavelengths (infrared) but predicted that as $\lambda \to 0$ (UV region), Energy $\to \infty$ . This absurd result is known as the Ultraviolet Catastrophe.

Planck’s Quantum Hypothesis (The Solution)

Max Planck (1900) proposed that energy exchange is not continuous but discrete.

Oscillators emit/absorb energy in packets called quanta (photons).
Energy of a photon: $E = nh\nu$ (where $n$ is an integer, $h$ is Planck's constant, $\nu$ is frequency).
Planck’s Radiation Law: accurately fits the experimental curve for all wavelengths.

3. Photoelectric Effect

Definition: The phenomenon of emission of electrons from a metal surface when electromagnetic radiation (light) of suitable frequency is incident on it.

Experimental Observations (Classical Failures):

Threshold Frequency: No emission occurs if the incident frequency $\nu < \nu_0$ (threshold frequency), regardless of intensity.
Instantaneous Process: There is no time lag between irradiation and emission.
Kinetic Energy: The maximum Kinetic Energy ( $K_{max}$ ) of emitted electrons depends on frequency, not intensity.
Saturation Current: Photoelectric current depends on intensity, provided $\nu > \nu_0$ .

Einstein’s Photoelectric Equation:
Einstein applied Planck's quantum theory. Light consists of photons of energy $E=h\nu$ . When a photon hits an electron, energy is used in two parts:

To overcome the surface barrier (Work Function, $\phi$ or $W$ ).
The remainder becomes Kinetic Energy.

h\nu = \phi + K_{max}

K_{max} = h\nu - h\nu_0 \quad (\text{Since } \phi = h\nu_0)

\frac{1}{2}mv^2 = h(\nu - \nu_0)

4. Concept of De Broglie Matter Waves

Hypothesis: Louis de Broglie (1924) proposed the hypothesis of Wave-Particle Duality.

Nature loves symmetry. If radiation (waves) can act like particles (Photoelectric effect), then matter (particles) should have wave-like properties.
These waves associated with moving material particles are called Matter Waves or de Broglie Waves.

De Broglie Wavelength ( $\lambda$ ):
Combining Planck’s energy equation ( $E = h\nu$ ) and Einstein’s mass-energy relation ( $E = mc^2$ ):

h\nu = mc^2 \implies \frac{hc}{\lambda} = mc^2

\lambda = \frac{h}{mc} = \frac{h}{p}

For a material particle moving with velocity

v

\lambda = \frac{h}{mv}

5. Wavelength of Matter Waves in Different Forms

A. In terms of Kinetic Energy ( $E$ )

Let $E$ be the kinetic energy of a particle of mass $m$ .

E = \frac{1}{2}mv^2 = \frac{p^2}{2m} \implies p = \sqrt{2mE}

\lambda = \frac{h}{\sqrt{2mE}}

B. For a Charged Particle Accelerated by Potential $V$

If a particle with charge $q$ is accelerated through a potential difference $V$ , the work done equals kinetic energy ( $E = qV$ ).

\lambda = \frac{h}{\sqrt{2mqV}}

C. Specific Case: For an Electron

Mass ( $m$ ) = $9.1 \times 10^{-31}$ kg
Charge ( $q$ ) = $1.6 \times 10^{-19}$ C
Planck's constant ( $h$ ) = $6.626 \times 10^{-34}$ Js
Substitute these values into the formula above:
$\lambda = \frac{12.27}{\sqrt{V}} \, \text{\AA}$

D. For Thermal Neutrons / Gas Molecules

For a gas particle at absolute temperature $T$ , Kinetic Energy $E = \frac{3}{2}k_BT$ (where $k_B$ is Boltzmann's constant).

\lambda = \frac{h}{\sqrt{3mk_BT}}

6. Phase Velocity and Group Velocity

In quantum mechanics, a single wave cannot represent a localized particle because a pure sine wave extends from $-\infty$ to $+\infty$ . Instead, a particle is represented by a Wave Packet (a group of waves with slightly different frequencies superimposing).

Phase Velocity ( $v_p$ )

Also called Wave Velocity.
It is the velocity with which a single monochromatic wave (a constant phase point) travels.
Formula: $v_p = \frac{\omega}{k} = \frac{E}{p}$
Qualitative Issue: For a particle, derivation shows $v_p = c^2/v$ . Since $v < c$ , $v_p > c$ . This means phase velocity can exceed the speed of light, carrying no physical information.

Group Velocity ( $v_g$ )

The velocity with which the "envelope" or the "packet" of waves moves.
It represents the velocity of energy transport.
Formula: $v_g = \frac{d\omega}{dk}$
Relation to Particle: It can be proven that the group velocity of the matter wave equals the actual velocity of the particle ( $v_g = v_{particle}$ ).

Relation between $v_g$ and $v_p$ :

v_g = v_p - \lambda \frac{dv_p}{d\lambda}

In a non-dispersive medium, $v_g = v_p$ .
In a dispersive medium (like matter waves), $v_g \neq v_p$ .

7. Heisenberg Uncertainty Principle

Statement: It is impossible to determine simultaneously both the exact position and the exact momentum of a microscopic particle.

Mathematical Formulation:
The product of the uncertainty in position ( $\Delta x$ ) and the uncertainty in momentum ( $\Delta p$ ) is always greater than or equal to a constant ( $\hbar/2$ ).

\Delta x \cdot \Delta p \ge \frac{\hbar}{2}

(Where $\hbar = \frac{h}{2\pi}$ )

Also typically written as: $\Delta x \cdot \Delta p \ge \frac{h}{4\pi}$

Other forms of Uncertainty:

Energy-Time Uncertainty: $\Delta E \cdot \Delta t \ge \frac{\hbar}{2}$
Angular Momentum-Angle: $\Delta L \cdot \Delta \theta \ge \frac{\hbar}{2}$

Significance:

Rejection of Trajectories: Electrons do not follow precise paths (orbits) as described by Bohr; they exist in "clouds" or probability zones.
Non-existence of electron in nucleus: Using the confinement of the nucleus size ( $\Delta x \approx 10^{-14}m$ ), calculations show the required electron velocity would exceed $c$ , proving electrons cannot exist inside a nucleus.

8. Wave Function and its Significance

Wave Function ( $\Psi$ ):
A mathematical function ( $\Psi(x, t)$ ) that describes the quantum state of a system. It contains all the information that can be known about the particle.

Physical Significance (The Born Interpretation):

$\Psi$ : The wave function itself has no direct physical meaning (it can be complex/imaginary).
$|\Psi|^2$ : The square of the absolute magnitude represents the Probability Density.
- $P(x) = |\Psi|^2 = \Psi \cdot \Psi^*$
- It represents the probability of finding the particle at a specific position at a specific time.

Characteristics of a Well-Behaved Wave Function:
For $\Psi$ to be physically acceptable, it must satisfy:

Finite: Must not go to infinity anywhere.
Single-valued: Only one probability for a given position.
Continuous: $\Psi$ and its first derivative $d\Psi/dx$ must be continuous.
Normalizable: The particle must exist somewhere in space.
$\int_{-\infty}^{+\infty} |\Psi|^2 \, dx = 1$

9. Schrödinger Equation

The fundamental equation of non-relativistic Quantum Mechanics (analogous to $F=ma$ in Classical Mechanics).

A. Time-Dependent Schrödinger Equation (STDE)

Describes how the wave function evolves over time.

i\hbar \frac{\partial \Psi}{\partial t} = \hat{H}\Psi

Expanding the Hamiltonian operator

\hat{H}

for 1D:

i\hbar \frac{\partial \Psi}{\partial t} = -\frac{\hbar^2}{2m} \frac{\partial^2 \Psi}{\partial x^2} + V(x,t)\Psi

$i$ : Imaginary unit
$\hbar$ : Reduced Planck's constant
$V$ : Potential energy

B. Time-Independent Schrödinger Equation (STIE)

Used for stationary states where Potential $V$ depends only on position $x$ , not time $t$ . We separate variables $\Psi(x,t) = \psi(x)\phi(t)$ .

The Equation (1D form):

\frac{d^2\psi}{dx^2} + \frac{2m}{\hbar^2}(E - V)\psi = 0

$E$ : Total Energy
$V$ : Potential Energy
$\psi$ : Spatial part of wave function

10. Particle in a 1D Box (Infinite Potential Well)

Problem Description:
Consider a particle of mass $m$ moving in a one-dimensional box of length $L$ with rigid walls.

Inside the box ( $0 < x < L$ ): The particle is free, so Potential $V = 0$ .
Outside the box ( $x \le 0$ or $x \ge L$ ): Potential $V = \infty$ (Particle cannot escape).

Derivation:

Wave Equation: Inside the box ( $V=0$ ), the STIE becomes:
$\frac{d^2\psi}{dx^2} + \frac{2mE}{\hbar^2}\psi = 0$
Let $k^2 = \frac{2mE}{\hbar^2}$ . Then, $\frac{d^2\psi}{dx^2} + k^2\psi = 0$ .
General Solution:
$\psi(x) = A\sin(kx) + B\cos(kx)$
Boundary Conditions:
- At $x=0$ , $\psi(0) = 0 \implies B = 0$ .
- At $x=L$ , $\psi(L) = 0 \implies A\sin(kL) = 0$ .
  Since $A \neq 0$ (particle must exist), $\sin(kL) = 0$ .
  $kL = n\pi \implies k = \frac{n\pi}{L} \quad (\text{where } n = 1, 2, 3...)$

Results:

1. Eigenvalues (Quantized Energy Levels):
Substitute $k = \frac{n\pi}{L}$ back into $k^2 = \frac{2mE}{\hbar^2}$ :

\frac{n^2\pi^2}{L^2} = \frac{2mE_n}{\hbar^2}

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

Energy is quantized ( $n=1,2,3...$ ).
$n=0$ is not allowed.

2. Zero Point Energy:
For $n=1$ (Ground state):

E_1 = \frac{h^2}{8mL^2}

Unlike classical mechanics, the lowest energy is non-zero. The particle is never at absolute rest.

3. Eigenfunctions (Wave Functions):
Using the normalization condition $\int_0^L |\psi|^2 dx = 1$ , we find $A = \sqrt{\frac{2}{L}}$ .

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