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**Unformatted text preview: **ket Notation
| If , | | flipped and starred, i.e.
transpose and conjugate ∗ | ∗ expression equals same expression
starred – only true if real 4 9/16/2013 Postulate 2 pq qp commutator h
2 i from original paper
Zeitschrift für Physik,
43 (1927), 172-198 multiply by 1 in the form of i/i ̂,
In classical physics, p and q are variables and the commutator is zero
In QM, they are operators as indicated with the “hats”. If we
choose
, then ̂
Whenever the operators of two variables do not commute (don’t equal zero),
then an uncertainty principle applies ̂,
̂, pick x-component ̂ ̂ ̂, ̂ hard to work with operators by
themselves, operate on an arbitrary
function ̂ ̂ ̂ ̂
̂ operators operate from right to left ̂ assume position representation ̂ see what happens when ̂ differentiate by parts
1st and 3rd term cancel satisfies commutator relation Whenever the operators of two variables do not commute (don’t equal zero),
, then
then an uncertainty principle applies, such as ̂ , xp x 2 (1.26) How do we get the uncertainties?
• choose form for position operator, get operator for momentum
• with operators, calculate expectation values for <x>,<x2>,<p>,<p2>
• uncertainties are x x2 x 2 p p2 p 2 5 9/16/2013 from Postulate 2 Operators, so far …
x-component of
kinetic energy position
x-component of
momentum ̂ 2 potential energy i energy (Hamiltonian)
x-component of
momentum squared 2
̂ Postulate 3
In any measurement of the observable
ˆ
associated with the operator A , the only
values that will ever be observed are the
eigenvalues a n , which satisfy the
ˆ
eigenvalue equation A n an n
Particle-in-a-Box Example
energy is so
important in QM, that
the operator for
time independent Schrödinger Eq.
energy has it’s own
name, the
,
1,2, …
Hamiltonian
wavefunctions or eigenfunctions
eigenvalues ( x) 2 nx sin 0 x a n 1,2,...
a a Orthonormal set of functions for solving problems with
other operators, , with Postulate 4
∗ Postulate 4
If a system is in a state described by a normalized wave
function , then th...

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