1. A QUANTUM LEAP IN MATH
By Michael Smith

2. Introduction to Quantum Mechanics
Niels Bohr
Erwin Schrödinger
Werner Heisenberg

3. 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

4. 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

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

6. 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 eV, -3.4 eV, and -1.5 eV agreed with experimental data

7. 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

8. 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

9. 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

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

11. 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

12. 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

13. 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

14. 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

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

16. 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”

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

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

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

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

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

22. Heisenberg’s Uncertainty PrincipleOr