Unit 4 - Notes

PHY110

Unit 4: Quantum Mechanics

1. Need for Quantum Mechanics

Classical mechanics (Newtonian mechanics) successfully explains the motion of macroscopic bodies (planets, projectiles, cars). However, toward the end of the 19th century, it failed to explain phenomena involving microscopic particles (electrons, atoms, molecules) and the interaction of radiation with matter.

Key Failures of Classical Mechanics:

Black Body Radiation: Classical theory (Rayleigh-Jeans Law) predicted that a black body should emit infinite energy at high frequencies (Ultraviolet Catastrophe), which contradicted experimental evidence.
Photoelectric Effect: Classical wave theory suggested that light intensity determines electron emission energy, while experiments showed it depended on frequency.
Atomic Stability: According to classical electromagnetism, an accelerating electron in an orbit should radiate energy and spiral into the nucleus. Atoms, however, are stable.
Atomic Spectra: Classical physics predicted continuous spectra for atoms, whereas hydrogen and other atoms exhibit discrete line spectra.

Conclusion: A new framework—Quantum Mechanics—was required to treat energy and matter as discrete (quantized) rather than continuous.

2. Photoelectric Effect

The phenomenon of the emission of electrons from a metal surface when electromagnetic radiation (light) of a suitable frequency falls on it.

Experimental Observations:

Instantaneous Process: Emission occurs immediately (time lag $< 10^{-9}$ s).
Threshold Frequency ( $\nu_0$ ): No emission occurs if the incident frequency is below a specific value characteristic of the metal, regardless of intensity.
Kinetic Energy: The maximum kinetic energy of emitted electrons depends linearly on the frequency of light, not intensity.
Saturation Current: The number of photoelectrons (current) is proportional to the intensity of incident light.

Einstein’s Explanation (Quantum Theory of Light):
Einstein proposed that light consists of discrete packets of energy called photons or quanta.

Energy of a photon: $E = h\nu$
Where $h = 6.626 \times 10^{-34} \text{ J}\cdot\text{s}$ (Planck’s constant).

Einstein’s Photoelectric Equation:
When a photon hits an electron, its energy is used in two ways:

To overcome the surface barrier (Work Function, $\Phi$ ).
To impart kinetic energy to the electron.

h\nu = \Phi + K_{max}

Where:

$\Phi = h\nu_0$ (Work function)
$K_{max} = \frac{1}{2}mv^2_{max}$ (Maximum Kinetic Energy)

Therefore:

K_{max} = h(\nu - \nu_0)

3. Concept of de Broglie Matter Waves

In 1924, Louis de Broglie proposed the hypothesis of Wave-Particle Duality.

Hypothesis: If electromagnetic waves (light) can behave like particles (photons), then moving material particles (like electrons) should exhibit wave-like properties.
These waves associated with material particles are called Matter Waves or de Broglie Waves.

de Broglie Wavelength Formula

For a photon: $E = h\nu = hc/\lambda$ and $E = mc^2$ (mass-energy equivalence).
Equating them implies momentum $p = mc = h/\lambda$ .

By analogy, for a material particle of mass $m$ moving with velocity $v$ :

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

Wavelength of Matter Waves in Different Forms

1. In terms of Kinetic Energy ( $E$ or $K$ ):
Since $E = \frac{p^2}{2m} \implies p = \sqrt{2mE}$

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

2. In terms of Accelerating Potential ( $V$ ) for a charged particle:
If a charge $q$ is accelerated through a potential difference $V$ , the work done equals kinetic energy ( $E = qV$ ).

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

3. specifically for an Electron:
Mass ( $m_e$ ) = $9.1 \times 10^{-31}$ kg, Charge ( $e$ ) = $1.6 \times 10^{-19}$ C.
Substituting these values:

\lambda = \frac{12.27}{\sqrt{V}} \AA

4. For a Gas Molecule (Thermal Equilibrium):
At temperature $T$ , average kinetic energy $E = \frac{3}{2}k_B T$ (where $k_B$ is Boltzmann constant).

\lambda = \frac{h}{\sqrt{3m k_B T}}

4. Heisenberg Uncertainty Principle

Werner Heisenberg stated that it is physically impossible to measure certain pairs of conjugate properties simultaneously with arbitrary precision.

Statement:
The product of the uncertainty in position ( $\Delta x$ ) and the uncertainty in momentum ( $\Delta p$ ) of a particle can never be less than $\frac{\hbar}{2}$ (where $\hbar = \frac{h}{2\pi}$ ).

Mathematical Form:

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

(Note: Some texts use $\Delta x \cdot \Delta p \geq h$ , but $\hbar/2$ is the rigorous lower bound).

Other Forms:

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

Significance:

It rules out the concept of a definitive trajectory (orbit) for electrons (Bohr model).
It explains why electrons cannot exist inside the nucleus (the confinement energy would be too high).
It is a fundamental property of nature, not a result of instrumental error.

5. Concept of Phase Velocity and Group Velocity

Since a single wave extends over all space, a localized particle corresponds to a Wave Packet (a group of waves of slightly different frequencies interfering constructively).

Phase Velocity ( $v_p$ )

The velocity with which a single monochromatic wave (a single phase) propagates.
Formula: $v_p = \frac{\omega}{k} = \frac{E}{p}$
For matter waves, $v_p = \frac{c^2}{v}$ (Since $v < c$ , $v_p > c$ ).
Significance: The phase velocity of a matter wave exceeds the speed of light, which carries no physical information.

Group Velocity ( $v_g$ )

The velocity with which the "envelope" or the wave packet moves.
This represents the velocity of energy transport or the particle itself.
Formula: $v_g = \frac{d\omega}{dk}$
Relation to Particle Velocity: For a non-relativistic 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): $v_g \neq v_p$ .

6. Wave Function and its Significance

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

Physical Significance (Max Born Interpretation):

The wave function $\Psi$ itself has no direct physical meaning (it can be a complex number).
The square of the absolute magnitude, $|\Psi|^2 = \Psi \cdot \Psi^*$ , represents the Probability Density.
$|\Psi(x,t)|^2 dx$ gives the probability of finding the particle in the region $x$ to $x+dx$ at time $t$ .

Conditions for a Well-Behaved Wave Function:
To be physically acceptable, $\Psi$ must be:

Finite: Everywhere (to avoid infinite probability).
Single-valued: Only one probability for a given position.
Continuous: $\Psi$ and its first derivative $\frac{\partial \Psi}{\partial x}$ must be continuous.
Normalizable: The total probability of finding the particle somewhere in space must be 1.
$\int_{-\infty}^{+\infty} |\Psi|^2 d\tau = 1$

7. Schrödinger's Wave Equations

Erwin Schrödinger developed the fundamental equation of non-relativistic quantum mechanics, describing how the wave function evolves.

A. Schrödinger Time-Dependent Equation (STDE)

Used when the potential energy $V$ changes with time or to describe the evolution of a state.
Derived from Total Energy $E = K + V$ using energy and momentum operators ( $\hat{E} \to i\hbar\frac{\partial}{\partial t}$ and $\hat{p} \to -i\hbar\nabla$ ).

The Equation (1D):

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

Or in operator form: $\hat{E}\Psi = \hat{H}\Psi$ (where $\hat{H}$ is the Hamiltonian operator).

B. Schrödinger Time-Independent Equation (STIE)

Used for stationary states where the potential $V$ is a function of position only (independent of time). The wave function can be separated into spatial and temporal parts: $\Psi(x,t) = \psi(x)e^{-iEt/\hbar}$ .

The Equation (1D):

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

Where:

$E$ = Total Energy
$V$ = Potential Energy
$m$ = Mass of particle

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

This is the simplest application of the Schrödinger equation.

The Setup:
Consider a particle of mass $m$ moving in a one-dimensional box of length $L$ .

Inside the box ( $0 < x < L$ ): The particle is free, so Potential $V = 0$ .
Outside the box ( $x \leq 0, x \geq L$ ): The walls are rigid, so Potential $V = \infty$ .

Boundary Conditions:
The particle cannot exist outside the box.

$\psi(0) = 0$
$\psi(L) = 0$

Solution:
Solving the STIE for $V=0$ :

\frac{d^2\psi}{dx^2} + \frac{2mE}{\hbar^2}\psi = 0

General solution:

\psi(x) = A \sin(kx) + B \cos(kx)

, where

k = \sqrt{\frac{2mE}{\hbar^2}}

Applying boundaries:

$\psi(0) = 0 \implies B = 0$ .
$\psi(L) = A \sin(kL) = 0$ .
Since $A \neq 0$ (particle must exist), $\sin(kL) = 0 \implies kL = n\pi$ (where $n = 1, 2, 3...$ ).

Results:

1. Eigenvalues (Quantized Energy Levels):
Substituting $k = \frac{n\pi}{L}$ into $E = \frac{\hbar^2 k^2}{2m}$ :

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

Energy is discrete (quantized).
$n=1$ is the Ground State.
Zero Point Energy: $E_1 \neq 0$ . The particle is never at rest.

2. Eigenfunctions (Wave Functions):
Using normalization ( $\int_0^L |\psi|^2 dx = 1$ ):

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

9. Tunneling Effect (Qualitative Idea)

The Scenario:
Consider a particle with energy $E$ approaching a potential barrier of height $V_0$ , where $E < V_0$ .

Classical Prediction:
The particle does not have enough energy to overcome the barrier. It should be totally reflected ( $R=1, T=0$ ). It cannot exist inside the barrier.

Quantum Prediction (Tunneling):
According to Schrödinger’s equation, the wave function inside the barrier ( $V_0 > E$ ) becomes an exponential decay function rather than becoming zero instantly.

\psi_{barrier} \propto e^{-kx}

If the barrier is sufficiently thin (width $a$ is small), the wave function $\psi$ will have a non-zero value on the other side of the barrier.

This means there is a finite probability that the particle will tunnel through the barrier despite having insufficient energy to climb over it.

Key Characteristics:

The transmission probability depends exponentially on the barrier width and the square root of the energy deficit ( $V_0 - E$ ).
It is a purely quantum phenomenon.

Applications:

Alpha Decay: Alpha particles tunnel out of the nucleus.
Scanning Tunneling Microscope (STM): Uses electron tunneling to image surfaces at the atomic level.
Tunnel Diode: A semiconductor device utilizing tunneling for high-speed switching.