The Schrodinger equation for matter waves Ψ(x,t) in one dimension: Basically just conservation of energy!

The Schrödinger equation in 1D

Mass of particle

Potential Complex i, with i2 = -1 space and time

coordinates

KE PE Etot

Simplification #1 when V(x) only (or V(x,y,z) in 3D):

Ψ(x,t) separates into position part dependent part ψ(x) and time dependent part Φ(t) =exp(-iEt/ħ). Ψ(x,t)= ψ(x)Φ(t)

Plug in, get equation for ψ(x) Seems like a good exercise for HW ;)

Same as in the book with V(x) for U(x)

“Time independent Schrödinger equation”

Most physical situations (e.g. H atom) no time dependence in V!

(Will use this eqn. in all Schrödinger Eq’n problems in this class!)

Time independent Schrodinger Equation

(Basically just conservation of Energy) KE + PE = E

1. Figure out what V(x) is, for situation given. 2. Guess or look up functional form of solution. 3. Plug in to check if ψ’s, and all x’s drop out, leaving equation

involving only bunch of constants; showing that trial solution is correct functional form.

4. Figure out what boundary conditions must be to make sense physically.

5. Figure out values of constants to meet boundary conditions and normalization

6. Multiply by time dependence Φ(t) =exp(-iEt/ħ) to have full solution if needed. Ψ(x,t) STILL HAS TIME DEPENDENCE!

|ψ(x)|2dx =1 -∞

∞

time independent eq.

From last class: Time dependent Schrödinger equation in 1D:

Time dependent Schrödinger equation in 3D (Cartesian):

Solving Schrod. equ. - simplest case (free space)

Electron in free space, no electric fields or gravity around. 1. What is V(x)? 2. Where does it want to be? 3. What are boundary conditions on ψ(x)?

1. constant. 2. no preference- all x the same.

3. none, could be anywhere.

Smart choice: constant V(x) V(x) ≡ 0!

Example: simplest case, free space V(x) = const.

Smart choice: constant V(x) V(x) ≡ 0!

Solution:

with:

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ψ(x) = Bsinkx, or:

No boundary conditions not quantized!

, or: any linear comb. of these

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ψ(x) = Aeikx

So almost have solution, but we still have to include time dependence in Ψ:

A bit of algebra, and identity eiθ = cosθ + isinθ gives:

k, and therefore E, can take on any value.

So we found the solution for the time-independent Schrödinger equation (special case, V(x) = 0 ):

, ,

(Solution of time-independent Schrödinger eqn. with V(x)=0)

, with

If V=0, then E = Kinetic energy.

So first term in Schröd. Eq. is always just kinetic energy!

Which free electron has more kinetic energy?

big k = big KE

small k = small KE

Curvature ∝ KE. Bending ψ tighter = more KE

a.

b.

x

x

A. a, B. b, C. both the same KE

k = 2π/λ

All this is so much fun!! (Especially the math…)

But why bother?

When is small ‘small’?

(When does quantum mechanics matter?)

Does it matter here? How to find out?

Moore’s Law

Why does Moore’s Law work?

Why can we get about twice the computing power every other year for the same amount of money?

Intel and AMD (and many others) can pack about twice the number of transistors on the same chip-area with every new chip-generation.

A reduction of the structure size by leads to twice the number of transistors on the same area for about the same cost. Nice! But wait!... (Intel: 2003: 90nm, 2005: 65nm, 2007: 45nm, 2009: 32nm, 2011: 22nm … ?)

The end of Moore's law? As feature density goes up, device and line sizes must get smaller and smaller. Semiconductor chips are made

with optical lithographic techniques.

Current feature sizes are ~22 nm. These are essentially at the diffraction limit for optical techniques using UV light. Problem making features much smaller.

As device sizes get smaller and smaller, then intrinsic quantum effects will get more and more important. This may be good or bad. Quantum dots, nanotubes, and all of “nanotechnology”.