Quantum mechanics and the Schrodinger equation notes — Unit 3

Classical mechanics (Newtonian mechanics and Maxwell's electromagnetism) successfully explained macroscopic phenomena like the motion of planets and classical optics. However, it completely failed at the microscopic (atomic) level.

⇒Phenomena Classical Physics Couldn't Explain:

• Black Body Radiation: Ultraviolet catastrophe.
• Photoelectric Effect: Why energy depends on frequency, not intensity.
• Compton Effect: The scattering of X-rays by electrons.
• Atomic Spectra: Why atoms emit discrete, quantized lines of light rather than a continuous spectrum.

Max Planck (1900) proposed a revolutionary idea to explain Black Body Radiation: Energy is not emitted or absorbed continuously. Instead, it is exchanged in discrete packets or "quanta."

E = hν = hc/λ

Where:
h = Planck's constant (6.626 × 10⁻³⁴ J·s)
ν = frequency of radiation
c = speed of light

This quantized energy exchange laid the absolute foundation for all of quantum mechanics.

When light of a sufficiently high frequency falls on a metal surface, electrons are ejected. This is the photoelectric effect.

⇒Einstein's Explanation (1905)

Einstein used Planck's theory, stating that light consists of particles called photons. When a photon hits an electron, it transfers all its energy ($h\nu$) instantly.

hν = Φ + K_max

Φ = Work Function (minimum energy to eject electron)
K_max = Maximum kinetic energy of ejected electron

When X-rays (high energy photons) are scattered by loosely bound electrons, the scattered X-rays have a longer wavelength (lower energy) than the incident X-rays. This proves light has particle-like momentum.

Arthur Compton treated this as an elastic collision between a photon and an electron, applying conservation of energy and momentum.

Δλ = λ' - λ = (h / m₀c) * (1 - cosθ)

Where:
θ = scattering angle
m₀ = rest mass of electron
h / m₀c = Compton wavelength (2.426 × 10⁻¹² m)

Louis de Broglie (1924) proposed a symmetrical hypothesis: If light (a wave) can behave like particles (photons), then matter (particles) should also behave like waves.

These waves associated with moving material particles are called Matter Waves or de Broglie waves.

λ = h / p = h / (mv)

Where:
p = momentum of the particle
m = mass
v = velocity

For macroscopic objects, $m$ is huge, so $\lambda$ is unobservably small. For electrons, $\lambda$ is in the X-ray range.

• Lighter particles have longer matter wavelengths ($\lambda \propto 1/m$).
• Faster particles have shorter matter wavelengths ($\lambda \propto 1/v$).
• Matter waves are not electromagnetic waves. They require no charge to exist (even neutrons have matter waves).
• They represent the probability of finding the particle in space (Probability Waves).

⇒Wavelength of an Accelerated Electron

If an electron is accelerated through a potential difference $V$, its kinetic energy is $eV = p^2/2m$.

λ = h / √(2m·e·V) ≈ 12.27 / √V Ångströms

This was the first experimental proof of de Broglie's matter waves. They bombarded a Nickel crystal with a beam of electrons and measured the scattered intensity.

They observed a distinct diffraction peak (a wave property!) at an accelerating voltage of 54V and scattering angle of 50°.

⇒The Verification

• Theoretical (de Broglie): $\lambda = 12.27 / \sqrt{54} = 1.67$ Å
• Experimental (Bragg's Law): Using X-ray diffraction techniques on the electron beam, $\lambda = 1.65$ Å.

The perfect match proved beyond doubt that electrons act as waves.

Introduced by Werner Heisenberg in 1927, this principle states that it is fundamentally impossible to measure both the exact position and exact momentum of a microscopic particle simultaneously.

Δx · Δp ≥ h / (4π)
OR
Δx · Δp ≥ ħ / 2

This is not due to flawed instruments; it is a fundamental property of nature arising from the wave nature of matter. To locate a particle precisely (small $\Delta x$), you must use very short wavelength light, which imparts huge unpredictable momentum (large $\Delta p$) to the particle.

⇒Energy-Time Uncertainty

It is impossible to precisely measure the energy of a system in an infinitely short time interval.

ΔE · Δt ≥ ħ / 2

⇒Angular Position-Angular Momentum

ΔL · Δθ ≥ ħ / 2

A particle cannot be represented by a single monochromatic wave because a single wave extends to infinity (so $\Delta x = \infty$, $\Delta p = 0$). Instead, a particle is represented by a Wave Packet—a superposition of many waves.

For a material particle, the group velocity of its matter wave exactly equals the mechanical velocity of the particle ($v_g = v_{particle}$).

In classical mechanics, we use coordinates $(x,y,z)$ to track a particle. In quantum mechanics, we describe the state of a particle using a complex mathematical function called the Wave Function, $\Psi(x,y,z,t)$.

⇒Physical Significance (Born Interpretation)

$\Psi$ itself has no direct physical meaning (it can be complex). However, its absolute square $|\Psi|^2$ (which equals $\Psi \Psi^*$) represents the Probability Density.

Probability P = ∭ |Ψ|² dV

The probability of finding the particle somewhere in the universe must be 1. This gives the Normalization Condition: $\iiint |\Psi|^2 dV = 1$.

Not every mathematical function can be a valid wave function. To be physically acceptable, $\Psi$ must satisfy standard conditions:

• Finite: $\Psi$ must be finite everywhere (probability cannot be infinite).
• Single-valued: $\Psi$ must have only one value at a given point (a particle cannot have two different probabilities of being in the same place).
• Continuous: $\Psi$ and its first derivatives ($\partial\Psi/\partial x$) must be continuous everywhere.
• Normalizable: $\Psi$ must approach zero as $x \to \pm\infty$ so the total probability integral is finite.

In quantum mechanics, every observable physical quantity (momentum, energy) is replaced by a mathematical operator that acts on the wave function.

Erwin Schrödinger (1926) developed the fundamental equation of quantum mechanics. We can "derive" it using classical energy and quantum operators.

Total Energy = Kinetic Energy + Potential Energy

E = p²/2m + V

Multiply by $\Psi$ and substitute the quantum operators:

iħ (∂Ψ/∂t) = -(ħ²/2m)∇²Ψ + VΨ

This equation dictates how the wave function of a quantum system evolves over time.

If the potential energy $V$ does not depend on time (stationary states), we can separate the spatial and temporal parts of the wave function: $\Psi(x,t) = \psi(x) \phi(t)$.

Substituting this into the time-dependent equation and dividing by $\psi\phi$ gives the Time-Independent Schrodinger Equation (in 1D):

d²ψ/dx² + (2m/ħ²)(E - V)ψ = 0

In 3D, replacing $d^2/dx^2$ with the Laplacian $\nabla^2$:

∇²ψ + (2m/ħ²)(E - V)ψ = 0

This is the most famous application of Schrodinger's equation. Imagine a particle (like an electron) trapped in a one-dimensional potential well of width $L$ with infinitely high, impenetrable walls.

⇒Boundary Conditions:

• Inside the box ($0 < x < L$): Potential $V = 0$.
• At the walls ($x = 0$ and $x = L$): Potential $V = \infty$.
• Since the particle cannot exist where $V = \infty$, the wave function $\psi(x)$ must be zero at the walls.

Applying the Schrodinger equation inside the box:

d²ψ/dx² + (2mE/ħ²)ψ = 0
Let k² = 2mE/ħ²
d²ψ/dx² + k²ψ = 0

The general solution to the differential equation $d^2\psi/dx^2 + k^2\psi = 0$ is:

ψ(x) = A sin(kx) + B cos(kx)

Applying boundary conditions:

• At $x = 0$, $\psi = 0$ ⟹ $B = 0$. So $\psi(x) = A \sin(kx)$.
• At $x = L$, $\psi = 0$ ⟹ $A \sin(kL) = 0$. Since $A$ cannot be zero (no particle), $\sin(kL) = 0$.

Therefore, $kL = n\pi$, where $n = 1, 2, 3...$

k = nπ / L

We defined $k^2 = 2mE/\hbar^2$. Substituting $k = n\pi/L$:

(n²π²)/L² = 2mE / (h/2π)²
E_n = (n²h²) / (8mL²)

⇒Physical Consequences:

• Energy is Quantized: The particle can only have specific, discrete energy values ($E_1, 4E_1, 9E_1...$).
• Zero-Point Energy: The lowest possible state is $n=1$, where $E_1 = h^2/(8mL^2)$. The energy can NEVER be zero. The particle is never at rest!

To find the constant $A$ in $\psi_n(x) = A \sin(\frac{n\pi x}{L})$, we use the Normalization condition: $\int_0^L |\psi(x)|^2 dx = 1$.

A² ∫₀ᴸ sin²(nπx/L) dx = 1
A² (L/2) = 1  ⟹  A = √(2/L)

The final normalized wave function for a particle in a 1D box is:

ψ_n(x) = √(2/L) sin(nπx/L)

Plotting $|\psi|^2$ shows that in state $n$, there are $n$ "bumps" (antinodes) of maximum probability, separated by $n-1$ nodes where the probability is strictly zero.

⇒Unit 3 Mastery

• Can you write Einstein's photoelectric equation and define work function?
• State the de Broglie hypothesis and derive $\lambda$ for an electron accelerated by $V$ volts.
• Write the mathematical statement of Heisenberg's Uncertainty Principle.
• What is the Born interpretation of the wave function? What makes a wave function acceptable?
• Derive the time-independent Schrodinger equation.
• Solve the particle in a 1D box problem to find $E_n$ and $\psi_n(x)$.