A QUANTUM LEAP IN MATH By Michael Smith

Introduction to Quantum Mechanics Niels Bohr Erwin Schrödinger Werner Heisenberg

Bohr’s Model of Atom Max Planck and quanta of energy E = hυ Bohr wanted to explain how electrons orbit nucleus Theorized orbits of electrons quantized

Bohr’s Model of Atom Force associated with the charged particle = to centripetal force of a rotating electron Angular momentum constant, quantized, and related to Planck’s constant

Bohr’s Model of Atom Using some algebra Bohr found radius Plugging in appropriate values first 3 orbits of Hydrogen atom are 0.529 Å, 2.116 Å, and 4.761Å which matched experimental data

Bohr’s Model of Atom Total Energy of System Using a little algebra and formula for radius Bohr found First 3 energy levels for hydrogen calculated to be -13.6 eV, -3.4 eV, and -1.5 eV agreed with experimental data

Schrödinger’s Equation de Broglie proposed that matter, like light, could possess properties of both particles and waves Also said energy equations developed by Einstein and Planck were equal From this stated that wavelength could be determined by knowing a particle’s mass and velocity

Schrödinger’s Equation Schrödinger convinced by 2 colleagues (Henri and DeBye) to come up with wave equation to explain de Broglie’s concept To verify applied it to hydrogen atom putting it in spherical coordinates

Schrödinger’s Equation Schrödinger equation made up of angular component (θ, φ) and radial component (r) Solving for each component, the wave function for hydrogen at ground state

Schrödinger’s Equation Only radial component at ground state, Schrödinger equation for hydrogen Solving gives or -13.6 eV which agrees with Bohr’s model and experimental data

Heisenberg’s Uncertainty Principle states that simultaneous measurements of position and momentum, of a particle can only be known with no better accuracy than Planck’s constant, h, divided by four times π

Heisenberg’s Uncertainty Principle Wave function is a probability density function. Heisenberg reasoned that probability would be normally distributed, or Gaussian in nature given by expression

Heisenberg’s Uncertainty Principle Heisenberg inferred that the Gaussian distribution of the position coordinate, q, would be expressed by δq is the half-width of the Gaussian hump where the particle will be found uncertainty in position is given by

Heisenberg’s Uncertainty Principle Momentum distribution given by Substituting the function for ψ(q) gives Which through the magic of algebra can be rewritten as

Heisenberg’s Uncertainty Principle Since second exponential term not dependent on “q”, the expression can be rewritten as Letting so gives

Heisenberg’s Uncertainty Principle This integral is symmetric so can be rewritten as To solve this requires some math trickery first squaring both sides and selecting another “dummy variable”

Heisenberg’s Uncertainty Principle Then rewriting it as a double integral Combine exponential terms

Heisenberg’s Uncertainty Principle because y2 + x2 = r2, the integral can be rewritten in polar coordinates as Integrating over θ gives

Heisenberg’s Uncertainty Principle Letting u = (1/2)r2 and du = rdr gives And solving gives Or

Heisenberg’s Uncertainty Principle Comparison of the function for momentum with the probability density function of normal distribution gives Which simplifies to

Heisenberg’s Uncertainty Principle Getting delta terms on left side gives Remembering where Δq is standard deviation, the same applies for δp, giving

Heisenberg’s Uncertainty Principle Or