]> Lecture 1

Quantum Mechanics

1/14/2008

Stanford class with Leonard Susskind. First class of the semester.

 

Randomness, Classical vs. Quantum Mechanics

Quantum mechanics is based upon a different type of randomness that can’t be duplicated within Classical Mechanics. For example adding small random forces in CM in an attempt to duplicate observed QM randomness fails to conserve energy. QM always conserves energy.

 

The two-slit experiment

Send photos toward the slit one at a time (dim beam).

 

Classical vs. Quantum statistics:

If the electron is detected going through one slit, then the statistics add as in Classical mechanics. However, if no measurement of the passage of the electron implies interference, then Quantum statistics.

 

Deterministic systems

Consider a  3-state system which endlessly cycles among the three states.

This system is reversible. Quantum Mechanics is also reversible.

 

The two slit experiment fires the electrons back to the gun if the detector at the slit is removed.

 

Can you determine the kick to the slit mechanism by measuring its momentum? Answer:  not accurately enough to violate uncertainty principle.

Heisenberg Uncertainty Principle

Einstein discovered for light (photons):

Energy:          E=hf=ω MathType@MTEF@5@5@+=faaafaart1ev1aqat0uyJj1BTfMBaerbuLwBLnharmWu51MyVXgaruqqWbhBLbYqOfMBJvMC5bqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaaceWaaqaaaOqaaiaadweacqGH9aqpcaWGObGaamOzaiabg2da9iabl+qiOjabeM8a3baa@36C0@ where the angular frequency is ω=2πf MathType@MTEF@5@5@+=faaafaart1ev1aqat0uyJj1BTfMBaerbuLwBLnharmWu51MyVXgaruqqWbhBLbYqOfMBJvMC5bqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaaceWaaqaaaOqaaiabeM8a3jabg2da9iaaikdacqaHapaCcaWGMbaaaa@3553@

Momentum:   p= hf c = ω c MathType@MTEF@5@5@+=faaafaart1ev1aqat0uyJj1BTfMBaerbuLwBLnharmWu51MyVXgaruqqWbhBLbYqOfMBJvMC5bqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaaceWaaqaaaOqaaiaadchacqGH9aqpdaWcaaqaaiaadIgacaWGMbaabaGaam4yaaaacqGH9aqpdaWcaaqaaiabl+qiOjabeM8a3bqaaiaadogaaaaaaa@38DB@

By definition, for any wave:  c=λf MathType@MTEF@5@5@+=faaafaart1ev1aqat0uyJj1BTfMBaerbuLwBLnharmWu51MyVXgaruqqWbhBLbYqOfMBJvMC5bqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaaceWaaqaaaOqaaiaadogacqGH9aqpcqaH7oaBcaWGMbaaaa@33A9@

then  p= h λ MathType@MTEF@5@5@+=faaafaart1ev1aqat0uyJj1BTfMBaerbuLwBLnharmWu51MyVXgaruqqWbhBLbYqOfMBJvMC5bqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaaceWaaqaaaOqaaiaadchacqGH9aqpdaWcaaqaaiaadIgaaeaacqaH7oaBaaaaaa@33C8@

For Heisenberg’s microscope, see Schiff.

Vector spaces

The basic logic of classical physics is set theory.

            Has states, and a transition rule for the states.

            A specific state is selected from a set of states.

           

The states become vectors in Hilbert space in QM which is not a set but an infinite dimensional vector space.

 

You must know the math of vector spaces over the complex numbers “C”.

 

States are represented by a set of vectors represented by “kets” |a MathType@MTEF@5@5@+=faaafaart1ev1aqat0uyJj1BTfMBaerbuLwBLnharmWu51MyVXgaruqqWbhBLbYqOfMBJvMC5bqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaaceWaaqaaaOqaamaaEiaabeqaaiaadggaaiaawEa7caGLQmcaaaa@3298@  “ket vector”. The dual vector space also has vectors represented by “bras” b| MathType@MTEF@5@5@+=faaafaart1ev1aqat0uyJj1BTfMBaerbuLwBLnharmWu51MyVXgaruqqWbhBLbYqOfMBJvMC5bqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaaceWaaqaaaOqaamaaEeaabaGaamOyaaqabiaawMYicaGLhWoaaaa@3297@

This space is linear and closed since the following sum is also a vector,

            α|a+β|b=|c MathType@MTEF@5@5@+=faaafaart1ev1aqat0uyJj1BTfMBaerbuLwBLnharmWu51MyVXgaruqqWbhBLbYqOfMBJvMC5bqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaaceWaaqaaaOqaaiabeg7aHnaaEiaabeqaaiaadggaaiaawEa7caGLQmcacqGHRaWkcqaHYoGydaGhcaqabeaacaWGIbaacaGLhWUaayPkJaGaeyypa0Zaa4HaaeqabaGaam4yaaGaay5bSlaawQYiaaaa@3EBB@

where α MathType@MTEF@5@5@+=faaafaart1ev1aqat0uyJj1BTfMBaerbuLwBLnharmWu51MyVXgaruqqWbhBLbYqOfMBJvMC5bqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaaceWaaqaaaOqaaiabeg7aHbaa@30BB@  and β MathType@MTEF@5@5@+=faaafaart1ev1aqat0uyJj1BTfMBaerbuLwBLnharmWu51MyVXgaruqqWbhBLbYqOfMBJvMC5bqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaaceWaaqaaaOqaaiabek7aIbaa@30BD@  are complex numbers.