2 2

2, , , ,

2i x t x t V x t x t

t m x

12nE n

incidenttransmittedreflected

I II

Heisenberg’s Matrix Mechanics1924: de Broglie suggests particles are waves

Mid-1925: Werner Heisenberg introduces Matrix Mechanics•Semi-philosophical, it only considers observable quantities•It used matrices, which were not that familiar at the time•It refused to discuss what happens between measurements•In 1927 he derives uncertainty principles

Late 1925: Erwin Schrödinger proposes wave mechanics•Used waves, more familiar to scientists at the time•Initially, Heisenberg’s and Schrödinger’s formulations were competing•Eventually, Schrödinger showed they were equivalent; different descriptions which produced the same predictions

Both formulations are used today, but Schrödinger is easier to understand

The Free Schrödinger Equation1925: Erwin Schrödinger proposes wave mechanics•Peter Debye suggested to him he needed to find a wave equation for quantum mechanics•He hit on the idea of using complex waves

•The rest is history

•Starting point: Energy/Momentum relationship•Multiply by the wave function on the right•Use de Broglie relations to rewrite

•Use relationships for complex waves to rewrite with derivatives

2

2

pE

m

E

2

, ,2

pE x t x t

m

2

2, ,2

x t k x tm

p k

ikx

it

1k

i x

it

2 2

2, ,

2i x t x t

t m x

, expx t ikx i t

,x t

Sample Problem with Free Schrödinger

2 2

2, ,

2i x t x t

t m x

Show that the following expression satisfies the free Schrödinger equation, and find the constant A:

21, exp

Aixx t

tt

2 2 21 1 1exp exp exp

Aix Aix Aix

t t t t t tt t t

2 2 2

3/2 2

1 1exp exp

2

Aix Aix iAx

t t t tt

Sample Problem with Free Schrödinger (2)Show that the following expression satisfies the free

Schrödinger equation, and find the constant A: 21

, expAix

x ttt

2 21 2exp exp

Aix Aix Aix

x t tt t t

2 2 2 2 2 2

2 2

1 2 4exp exp

Aix Ai A i x Aix

x t tt t t t t

2 2

2, ,

2i x t x t

t m x

2 2 2 2 2

3/2 2 2

1 2 4

2 2

iAx Ai A i xi

t mt t t t t t

Multiply by 2mt5/2/

2 22 2 2m it Ax A it Ax

2

mA

The Schrödinger Equation

What if we have forces?•Need to add potential energy V(x,t) on top of kinetic energy term

2

2

pE

m ,V x t

2 2

2,

2i V x t

t m x

2 2

22i V

t m x

The General Prescription for Classical Quantum:1. Write a formula for the energy in terms of momentum and position2. Transform Energy and momentum using the following prescription:

3. Rewrite it as a wave equation

E it

p k

i x

2 2

2, , , ,

2i x t x t V x t x t

t m x

Comments on Schrödinger Equation1. This equation is inherentlycomplex•You MUST use complexwave functions2. This equation is first order in time•It has only first derivatives with respect to time•If you know the value at t = 0, you can work it out at subsequent times

•Proved using Taylor expansion:

2 2

2, , , ,

2i x t x t V x t x t

t m x

,x t 2

212 2

, , ,x t x t x tt t

, ,x t x tt

2 2

2, , , ,

2x t x t V x t x t

i m x

Initial conditions:Classical physics

x(t = 0) and v(t = 0)

Initial conditions:Quantum physics

(x,t = 0)

The Superposition Principle3. This equation is linear

•The wave functionappears to the first power everywhere•You can take linear combinations of solutions:

2 2

2, , , ,

2i x t x t V x t x t

t m x

Let 1 and 2 be two solutions of Schrödinger. Then so is

1 1 2 2, , ,x t c x t c x t where c1 and c2 are arbitrary

complex numbers

1 21 2

, , ,x t x t x ti c i c i

t t t

2 22 2

1 21 1 2 22 2

, ,, , , ,

2 2

x t x tc V x t x t c V x t x t

m x m x

2 2

1 1 2 2 1 1 2 22, , , , ,

2c x t c x t V x t c x t c x t

m x

22

2

,, ,

2

x tV x t x t

m x

Q.E.D

Time Independent problems•Often [usually] thepotential does notdepend on time: V = V(x).•To solve this equation, we try separation of variable:•Plug this guess in:•Divide by theoriginal wave function•Note that left side is independent of x, and right side is independent of t.

•Both sides must be independent of both x and t•Both sides must be equal to a constant, called E (the energy)

2 2

2, , ,

2i x t x t V x x t

t m x

,x t t x

i x tt

V x t x 2 2

22t x

m x

d ti

t dt

22

22

d xV x

m x dx

E

Solving the time equation

•We have turned one equation into two•But the two equations are now ordinary differential equations

•Furthermore, the first equation is easy to solve:

22

22

d t d xiE V x

t dt m x dx

d t Edt

t i

d iEdt

lniEt

t

iEtt e

, iEtx t e x

,x t t x

•These types of solutions are called stationary states•Why? Don’t they have time in them?

•The probability density is independent of time

2 2, *iEt iEtx t e x e x x

The Time Independent Schrödinger Eqn

22

22

d xE V x

m x dx

Multiply by (x) again

22

22

d xE x V x x

m dx

2 2

22

dE V

m dx

•This equation is much easier to solve than the

original equation•ODE’s are easier than PDE’s•It can pretty easily be solved numerically, if necessary•Note that it is a real equation – you don’t need complex numbers

•Imagine finding all possible solutions n(x) with energy En

•Then we can find solutions of the original Schrödinger Equation

, expn nx t x iE t •The most general solution is superposition of this solution

, expn n nn

x t c x iE t