Intro Lec9
to Quantum Mechanics
Numerical solution of Schdinger's equao tion
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Why/when quantum ?
Newton's laws give a very accurate description of the behavior of everyday objects/motions.
But they fail miserably to describe atoms !
This was a crisis for physics at turn of century ...
Eg Laws of EM + Newton mechanics predicts atoms should be unstable
Electrons classically have any energy but see only discrete energies
Energy of electromagnetic waves in vacuum infinite!
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Photoelectric effect light waves like particles !
Recent expts: eg. electron diffraction ...
Resolution
Radical. Took many physicists about 20 years to discover.
Discovered twice. Schrdinger, Heisenberg o (1926).
Arguably become the most well tested and accurate scientific theory (QED)
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Basics
Discard notion that microscopic objects like electrons can a well defined position, velocity etc. Not a practical issue but one of principle! Instead think of them as being described by a wavefunction (x,t). Like a usual wave in sense that electron is not localized like a classical particle. But this is a probability wave. |(x, t)|2 yields probability of finding particle at (x, t). Dynamics: replace Newton's laws (simple ordinary differential equations) by Schrdnger's o equation (partial differential equation)
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Points to note
2 2 h + V = i h - 2m x2 t Equation looks like a funny wave equation (but only first order in t)
Involves square root of minus 1 (i). In general is complex! Hence need ||2 for positive real probability
New fundamental introduced constant = h 1.05 10-34 Js. Planck's constant.
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Stationary states
h Put = (x)e-iEt/ . Plug into equation. Find time independent Schrdinger equao tion
2 2 h - + V = E 2m x2 Such a wavefunction describes the allowed state of say electron in an atom with E being its energy.
Try to solve schematically. In general only certain energies allowed!
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Allowed energies
Rewrite equation: d2 2m (V (x) - E ) = 2 dt2 h For a bound state need V > E at large x.
In which case this equation develops exponential solutions ex , e-x with 2 = 2m(V (x)-E) . 2
h
For small x V < E and generate oscillatory solutions sin x, cos x
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Allowed energies II
Need - dx2 (x) = 1. Probability.
Thus need to choose correct solution as x .
Inside will get oscillations.
Must smoothly match at boundary.
Requires that E be very carefully chosen. In general discrete set of possible E's.
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How to solve...