The Schrdinger Equation

1
The Schrödinger Equation
2
Quantization as an Eigenvalue Problem E.
Schrödinger
In this communication I wish to show, first for
the simplest case of the non-relativistic and
unperturbed hydrogen atom, that the usual rules
of quantization can be replaced by another
postulate, in which there occurs no mention of
whole numbers. Instead, the introduction of
integers arises in the same natural way as, for
example, in a vibrating string for which the
number of nodes is integral. The new conception
can be generalized and I believe that it
penetrates deeply into the true nature of the
quantum rules. translated in Introduction to
Quantum Mechanics by Linus Pauling and E. Bright
Wilson.
3
Eigenfunctions and Eigenvalues

• Eigen german for special, but also has a
  sense of belonging to in it.
• d2/d?2 (sin 3?) d/d? (3 cos 3?) -9 sin 3?

4
Operators

• In math we perform operations
• 2 5 The operator tells us we are to add
  the operands (2 and 5) together.
• Operators , -, /,x,v,sin, cos, ?, d/dx, d2/dx2

5
Observables

• Any phenomenon we can observe (and in particular
  measure) we call an observable
• To any observable we can associate an operator.

6
The Schrödinger Equation

• H? E?
• H is an operator.
• is a function.
• E is a number.

7
The Schrödinger Method

• Determine the operator associated with the
  observable to be calculated.
• Find the eigenvalues and eigenfunctions of the
  operator.
• Ensure that the eigenfunctions are physically
  acceptable, in which case the corresponding
  eigenvalues will be the possible, quantised
  values of the observable.

8
Electron on a ring

• Electron confined to velocity v in a circular
  path of radius r.
• Classical expression for the operator is
• E 1/2 m v2 (m v r)2/2m r2

9
Electron on a ring

• Create quantum mechanical operator for problem in
  the following way.
• See whiteboard..