lecture classicaltoquantum1 web

8/10/2019 Lecture ClassicaltoQuantum1 Web

1/7


2/7

Bohr assumed angular momentum, m r v, must be quantized ....

=

= (* *)

[{} / ]

= %[[ ]]

use cgs units (to not have to deal with permitivity of vacuum):

= /

= -

= -

=

= -

= -

/ - -(* /*)

Physically, this is the ...

Substitute this value of v,into the ....

2 lecture-ClassicaltoQuantum1-web.nb


3/7

[{ } / ]

define r:

= %[[ ]]

/ - - -(**)

Calculate total energy - Kinetic + Potential:

=

/

This is simply -1

2m velec

2, and is the ....

Importantly, this gets the ....

%

[ ]

= %[[ ]]

- / - - - (* -*)

-

=

(* -*)

lecture-ClassicaltoQuantum1-web.nb 3


4/7

/

We see that transitions between any two states should be the Rydberg constant times the difference in

inverse n2for each of the levels:

[_ _]=

( ( - ))- (*

*)



5/7

Our goal in this class was to learn how to specify the state of a quantum particle (electron, atom,molecule, etc.), then extract info from this knowledge.

How do we specify the state of a classical particle?

What's the distinction between quantum and classical??

What do we mean by state?

To answer these questions, let's first consider a simple physical system, the classical particle on a

spring - or Harmonic Oscillator

We have a particle attached to a massless, frictionless spring with force constant k.

The position-dependent force (magnitude and direction) is given by F

(x) = - kx

.



6/7

If we wanted to, we could use Newtons 2nd law (F

= ma

= m

2x

t2) to set up a differential equation for

x(t) and solve

F

a

x

t

[] -

[]

The syntax in Mathematicais to use double-prime ( ' ') to indicate second derivative, and the double-

equal sign indicates the definition of a function

The syntax is DSolve[differential equation, for some arbitrary function f(x), with independent variable x]

.

[{ } ]

So, the general solution, x[t], to Newton's equation of motion is ...

The frequencyis...

the coefficients would be determined by...

If we wanted to know the momentum of the particle at some time t, all we would have to do is find p(t) =

mx

t

[] =

[] =

So what does this tell us about the state of our particle?

two initial conditions, p[0], x[0] this gives the trajectory ata all times t, if we also know x[t]

It is thetrajectoryof the particle which defines the state of a classical particle, and it is specified com-

pletely by the following three things:



7/7

x[0], p[0] (initial position and momentum at time t = 0), and the HAMILTONIAN, H,for the system.

The Hamiltonian ...

heres how it works:

The Classical HAMILTONIANis a functionof position, x, and momentum, p, udefined as the SUM of

Kinetic (T) and Potential (V) Energies

H(x, p) = T(p) +V(x)

The kinetic energy, T =1

2mv2 or ....

The Potential energy is related to the Force:V

x= -Fx

And the differential equations of motion in the Hamiltonian picture (also referred to as the Hamiltonian

Equations of Motion) are

[]

=

()

[]

= -

= -

()

()

So, we see that Hamilton' s formulation and Newton' s are the same. For our purposes, momentum is a

more helpful description than is velocity or acceleration.

It is always the potential that is the tricky part.

Examples of different potentials are :

free particle - V =

harmonic oscillator - V=

These and other potentials will be solved quantum mechanically throughout the course.


lecture classicaltoquantum1 web

Documents