Quantum Physics
Table of Contents
- Birth of Quantum Mechanics
- Photoelectric Effect and Wave-Particle Duality
- Bohr Model of the Atom
- Wave Function, Superposition, and Wave Packets
- Uncertainty Principle
- Schrodinger Equation
- Expectation Value
Birth of Quantum Mechanics
Blackbody Radiation
A blackbody is an ideal object that:
- Absorbs 100% of incident electromagnetic radiation.
- Emits radiation that depends only on its temperature.
- Has an emission spectrum independent of its material.
A cavity with a small hole approximates a blackbody because radiation entering the hole undergoes many reflections and is almost completely absorbed before escaping.
Intensity and Spectral Intensity
Total intensity (power per unit area):
\[ I = \frac{E}{A t} \]
Spectral intensity (power per unit area per unit frequency):
\[ I_f = \frac{dE}{dA , dt , df} \]
Total intensity is obtained by integrating over all frequencies:
\[ I = \int_0^\infty I_f , df \]
Temperature Dependence
As temperature increases:
- The peak of the spectrum shifts to higher frequency.
- The peak wavelength decreases.
- The total emitted intensity increases rapidly.
Hotter objects change visible color:
- ~3000 K → red glow
- ~6000 K → yellow-white
- ~12000 K → bluish-white
Wien’s Displacement Law
Peak frequency form:
\[ f_{\text{peak}} = (5.88 \times 10^{10} , \text{Hz/K}) , T \]
Peak wavelength form:
\[ \lambda_{\text{peak}} = \frac{2.90 \times 10^{-3},\mathrm{m\cdot K}}{T} \]
Scaling behavior:
- If \(T\) doubles → \(f_{\text{peak}}\) doubles.
- If \(T\) doubles → \(\lambda_{\text{peak}}\) halves.
Stefan–Boltzmann Law
Total emitted intensity:
\[ I = \sigma T^4 \]
where
\[ \sigma = 5.67 \times 10^{-8},\mathrm{W,m^{-2},K^{-4}} \]
If temperature doubles, total emitted power increases by \(2^4 = 16\).
Classical Prediction: Rayleigh–Jeans Law
Classical equipartition assumes each mode has average energy:
\[ E = k_B T \]
Number of modes per unit frequency is proportional to \(f^2\).
Rayleigh–Jeans law:
\[ I_f^{RJ}(f,T) = \frac{2 f^2}{c^2} k_B T \]
Total energy prediction:
\[ \int_0^\infty f^2 , df = \infty \]
This divergence at high frequency is called the ultraviolet catastrophe.
Planck’s Quantum Hypothesis
Planck proposed that energy is quantized:
\[ E = n h f, \quad n = 0,1,2,\dots \]
where
\[ h = 6.626 \times 10^{-34},\mathrm{J\cdot s} \]
Average energy per mode becomes:
\[ \langle E \rangle = \frac{h f}{e^{hf/k_B T} - 1} \]
Planck’s radiation law:
\[ I_f(f,T) = \frac{2 f^2}{c^2} \frac{h f}{e^{hf/k_B T} - 1} \]
Why Planck’s Law Resolves the Catastrophe
At high frequency (\(hf \gg k_B T\)):
\[ e^{hf/k_B T} \gg 1 \]
Thus
\[ \langle E \rangle \rightarrow 0 \]
High-frequency modes are exponentially suppressed, preventing divergence.
Low-Frequency Limit (Classical Recovery)
When \(hf \ll k_B T\), use the approximation:
\[ e^{hf/k_B T} \approx 1 + \frac{hf}{k_B T} \]
Substituting:
\[ \langle E \rangle \approx k_B T \]
Planck’s law reduces to the Rayleigh–Jeans result at low frequencies.
Photon Energy
Energy of a single photon:
\[ E = h f \]
Using wavelength:
\[ E = \frac{hc}{\lambda} \]
Key Constants
Planck constant:
\[ h = 6.626 \times 10^{-34},\mathrm{J\cdot s} \]
Boltzmann constant:
\[ k_B = 1.381 \times 10^{-23} , \text{J/K} \]
Speed of light:
\[ c = 3.00 \times 10^8 , \text{m/s} \]
Stefan–Boltzmann constant:
\[ \sigma = 5.67 \times 10^{-8} , \text{W/m}^2\text{K}^4 \]
Conceptual Takeaways
- Classical physics fails because it assumes continuous energy.
- The number of modes grows with \(f^2\), causing divergence.
- Quantization suppresses high-frequency energy.
- Planck’s law matches experiment at all frequencies.
- At low frequency → classical behavior.
- At high frequency → quantum behavior dominates.
Photoelectric Effect and Wave-Particle Duality
The Photoelectric Effect
The photoelectric effect occurs when light shines on a metal surface and electrons are ejected.
Key experimental observations:
- No electrons are emitted below a certain threshold frequency.
- Increasing light intensity increases the number of emitted electrons.
- Increasing light frequency increases the kinetic energy of emitted electrons.
- Emission occurs instantly, even at low intensity.
These results could not be explained using classical wave theory.
Classical Prediction (Incorrect)
Classical wave theory predicted:
- Energy should depend on intensity, not frequency.
- Electrons should accumulate energy gradually.
- There should be a measurable time delay before emission.
Experiments showed this was completely wrong.
Einstein’s Explanation (1905)
Einstein proposed that light consists of particles called photons.
Each photon carries energy:
\[ E = h f \]
When a photon strikes an electron:
- Part of its energy is used to overcome the metal’s work function.
- The remainder becomes kinetic energy.
Photoelectric Equation
Energy conservation gives:
\[ h f = W + K_{\text{max}} \]
where:
- \(W\) = work function (minimum energy needed to remove electron)
- \(K_{\text{max}}\) = maximum kinetic energy of emitted electrons
Thus:
\[ K_{\text{max}} = h f - W \]
Threshold Frequency
The threshold frequency \(f0\) occurs when \(K{\text{max}} = 0\):
\[ h f_0 = W \]
So:
\[ f_0 = \frac{W}{h} \]
If \(f < f_0\), no electrons are emitted — regardless of intensity.
This was impossible under classical physics.
Experimental Graph
If we plot \(K_{\text{max}}\) vs frequency:
\[ K_{\text{max}} = h f - W \]
This is a straight line:
- Slope = \(h\)
- Intercept = \(-W\)
- Zero crossing = \(f_0\)
This experiment allowed direct measurement of Planck’s constant.
Intensity vs Frequency
- Intensity controls number of photons → number of electrons emitted
- Frequency controls energy per photon → kinetic energy of electrons
Do not confuse these.
Wave–Particle Duality
Blackbody radiation suggested light energy is quantized.
The photoelectric effect showed:
- Light behaves like discrete particles.
- Energy transfer is localized and instantaneous.
Thus light exhibits particle-like behavior.
But interference experiments show light also behaves like a wave.
Conclusion:
Light has wave–particle duality.
It cannot be described fully as only a wave or only a particle.
Momentum of a Photon
Photons carry momentum even though they have no mass.
Using:
\[ E = p c \]
Since \(E = h f\) and \(f = \frac{c}{\lambda}\):
\[ p = \frac{h}{\lambda} \]
This will later connect to de Broglie matter waves.
Key Constants
Planck constant:
\[ h = 6.626 \times 10^{-34},\mathrm{J\cdot s} \]
Electron rest mass:
\[ m_e = 9.11 \times 10^{-31},\mathrm{kg} \]
Speed of light:
\[ c = 3.00 \times 10^8,\mathrm{m,s^{-1}} \]
Conceptual Takeaways
- Light energy is quantized.
- Frequency determines photon energy.
- Intensity determines photon number.
- Wave and particle descriptions are both necessary.
Bohr Model of the Atom
Atomic Spectra
When atoms are excited, they emit light at discrete wavelengths.
Instead of a continuous spectrum, hydrogen produces sharp spectral lines.
These lines form series:
- Lyman series → ultraviolet
- Balmer series → visible
- Paschen series → infrared
This showed that atomic energy levels are discrete.
The Rydberg Formula
The wavelengths of hydrogen spectral lines follow the Rydberg formula:
\[ \frac{1}{\lambda} = R_H \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right) \]
where
- \(R_H = 1.097 \times 10^7 , \mathrm{m^{-1}}\)
- \(n_i\) = initial energy level
- \(n_f\) = final energy level
For emission:
\[ n_i > n_f \]
Bohr’s Model (1913)
Niels Bohr proposed a model of the hydrogen atom with three key assumptions:
- Electrons move in circular orbits around the nucleus.
- Only certain quantized orbits are allowed.
- Radiation is emitted only when electrons transition between energy levels.
Quantization of Angular Momentum
Bohr proposed that the electron’s angular momentum is quantized:
\[ m v r = n \hbar \]
where
- \(n = 1,2,3,\dots\)
- \(\hbar = \frac{h}{2\pi}\)
This condition restricts electrons to specific allowed orbits.
Radius of Allowed Orbits
Solving the force balance between Coulomb attraction and centripetal force gives:
\[ r_n = n^2 a_0 \]
where
\[ a_0 = 5.29 \times 10^{-11} , \text{m} \]
is the Bohr radius.
Energy Levels of Hydrogen
The allowed energies of hydrogen are:
\[ E_n = -\frac{13.6 , \text{eV}}{n^2} \]
Properties:
- Energy levels become closer together as \(n\) increases.
- \(E = 0\) corresponds to a free electron.
Photon Emission During Transitions
When an electron moves between levels:
\[ \Delta E = E_i - E_f \]
A photon is emitted with energy:
\[ hf = E_i - E_f \]
Thus:
\[ \lambda = \frac{hc}{E_i - E_f} \]
This explains the hydrogen spectral lines.
Connection to de Broglie Waves
The Bohr orbit condition can be interpreted as a standing wave condition:
\[ 2\pi r = n\lambda \]
Only wavelengths that fit an integer number of times around the orbit are allowed.
This links atomic structure to matter waves.
Conceptual Takeaways
- Atomic energy levels are quantized.
- Spectral lines arise from electron transitions.
- Angular momentum is quantized in units of \(\hbar\).
- Bohr’s model explains hydrogen spectra but fails for multi-electron atoms.
Wave Function, Superposition, and Wave Packets
The Wave Function
Quantum particles are described by a wave function:
\[ \psi(x,t) \]
The wave function contains all information about the particle.
The probability density of finding a particle is:
\[ P(x,t) = |\psi(x,t)|^2 \]
Normalization
Because the particle must exist somewhere:
\[ \int_{-\infty}^{\infty} |\psi(x,t)|^2 dx = 1 \]
This is called the normalization condition.
Plane Waves
A free particle can be described by a plane wave:
\[ \psi(x,t) = A e^{i(kx - \omega t)} \]
where
- \(k\) = wave number
- \(\omega\) = angular frequency
Relations:
\[ k = \frac{2\pi}{\lambda} \]
\[ p = \hbar k \]
\[ E = \hbar \omega \]
Superposition of Waves
Quantum wave functions obey the principle of superposition.
If \(\psi_1\) and \(\psi_2\) are valid solutions, then:
\[ \psi = \psi_1 + \psi_2 \]
is also a valid solution.
Interference between waves produces new spatial structures.
Superposition of Two Waves
Consider two waves:
\[ \psi_1 = A \sin(k_1 x) \]
\[ \psi_2 = A \sin(k_2 x) \]
Their sum is
\[ \psi = 2A \sin(k_{avg} x)\cos\left(\frac{\Delta k}{2}x\right) \]
This creates an envelope that localizes the wave.
Wave Packets
A wave packet is formed by combining many waves with different \(k\) values:
\[ \psi(x) = \int g(k)\cos(kx),dk \]
Wave packets represent localized particles.
Properties:
- Narrow packet → wide range of \(k\)
- Wide packet → narrow range of \(k\)
Spatial Localization
The spread in position and wave number satisfies
\[ \Delta x \Delta k \sim 1 \]
Using \(p = \hbar k\) gives the basis for the uncertainty principle.
Conceptual Takeaways
- The wave function describes probability amplitudes.
- Superposition allows interference and localization.
- A localized particle corresponds to a wave packet.
- Localization requires a range of momenta.
Uncertainty Principle
Position–Momentum Uncertainty
A wave packet cannot have both perfectly defined position and momentum.
The fundamental relation is
\[ \Delta x \Delta p \ge \frac{\hbar}{2} \]
This is the Heisenberg uncertainty principle.
Origin of the Uncertainty Relation
A localized particle requires many wave numbers.
From Fourier analysis:
\[ \Delta x \Delta k \approx \frac{1}{2} \]
Using
\[ p = \hbar k \]
we obtain
\[ \Delta x \Delta p \ge \frac{\hbar}{2} \]
Energy–Time Uncertainty
A similar relation exists for energy and time:
\[ \Delta E \Delta t \gtrsim \frac{\hbar}{2} \]
Short-lived states have large energy uncertainty.
Physical Interpretation
The uncertainty principle does not arise from measurement errors.
Instead it reflects a fundamental property of quantum states.
Key consequences:
- A particle cannot have a perfectly defined trajectory.
- Confinement increases momentum uncertainty.
Example: Particle in a Nucleus
If a proton is confined to
\[ \Delta x \sim 10^{-15} , \text{m} \]
then
\[ \Delta p \gtrsim \frac{\hbar}{2\Delta x} \]
This implies a large minimum kinetic energy.
Conceptual Takeaways
- Position and momentum cannot both be precisely known.
- Localization requires a wide momentum distribution.
- Quantum uncertainty is negligible for macroscopic objects.
Schrodinger Equation
Operators in Quantum Mechanics
Physical quantities are represented by operators.
Momentum operator:
$$ \hat{p} = -i\hbar \frac{\partial}{\partial x} $$
Energy operator:
$$ \hat{E} = i\hbar \frac{\partial}{\partial t} $$
Time-Dependent Schrödinger Equation
The fundamental equation of quantum mechanics is
\[ i\hbar \frac{\partial \psi}{\partial t} = -\frac{\hbar^2}{2m} \frac{\partial^2 \psi}{\partial x^2} + V(x),\psi \]
where
- \(m\) = particle mass
- \(V(x)\) = potential energy
Time-Independent Schrödinger Equation
If the potential does not depend on time:
\[ -\frac{\hbar^2}{2m} \frac{d^2 \psi}{dx^2} + V(x),\psi = E\psi \]
Solutions give the allowed energy levels.
Free Particle
For a free particle:
\[ V(x) = 0 \]
Solutions are plane waves:
$$ \psi = Ae^{ikx} + Be^{-ikx} $$
Energy:
$$ E = \frac{\hbar^2 k^2}{2m} $$
Particle in an Infinite Potential Well
Potential:
- \(V = 0\) inside the box
- \(V = \infty\) at the walls
Boundary conditions:
\[ \psi(0) = \psi(L) = 0 \]
Wave functions:
\[ \psi_n(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{n\pi x}{L}\right) \]
Allowed energies:
\[ E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2} = \frac{n^2 h^2}{8mL^2} \]
Quantum Tunneling
If a particle encounters a barrier with
\[ E < V_0 \]
classically it cannot cross.
Quantum mechanically, the wave function penetrates the barrier.
Transmission probability decreases exponentially with barrier width.
Conceptual Takeaways
- Schrödinger’s equation governs quantum dynamics.
- Boundary conditions produce quantized energies.
- Wave functions determine probability distributions.
- Quantum particles can tunnel through barriers.
Expectation Value
Probability Density
The probability density of finding a particle at position \(x\) is
\[ P(x) = |\psi(x)|^2 \]
The total probability must equal 1.
Expectation Value of Position
The expectation value of position is
\[ \langle x \rangle = \int_{-\infty}^{\infty} x |\psi(x)|^2 dx \]
This represents the average result of many measurements.
Expectation Value of an Operator
For any observable operator \(\hat{O}\):
\[ \langle O \rangle = \int \psi^* \hat{O} \psi , dx \]
Momentum Expectation Value
Using the momentum operator:
\[ \langle \hat{p} \rangle = \int \psi^* \left( -i\hbar \frac{\partial}{\partial x} \right) \psi , dx \]
Energy Expectation Value
Using the energy operator:
\[ \langle \hat{E} \rangle = \int \psi^* \left( i\hbar \frac{\partial}{\partial t} \right) \psi , dx \]
Expectation vs Measurement
Important distinctions:
- The expectation value is not the most likely value.
- It is the average over many measurements.
Individual measurements may produce different results.
Conceptual Takeaways
- Quantum predictions are probabilistic.
- The wave function determines measurement statistics.
- Expectation values correspond to measurable averages.