Classical Mechanics - sfgsfg.weebly.comsfgsfg.weebly.com/uploads/2/0/1/.../lecture_2_-_classical_mechanics... · Classical Mechanics. Classical Mechanics describes motion of particles.

Classical MechanicsClassical Mechanics describes motion of particles. Each particle has mass m and a well-defined position in space, e.g.

x - in 1D case (motion along a straight line)(x, y, z) - in 3D Cartesian coordinates (r, θ, φ) – in spherical polar coordinates

Classical trajectory: position as a function of time t.e.g. x(t) in 1D casex(t), y(t), z(t) in 3D Cartesian case

Classical trajectory contains a complete unambiguous knowledge about the motion of the system.

Classical trajectory can be found by solving the Equations of Motion (EOM)EOM are consequences of Physical Laws, e.g.

(1) 2nd Newton’s Law:

(2) Energy Conservation Law:

2

2

or dp Fdt

d x dVmdt dx

=

= −22

Mass

Momentum

1Kinetic Energy =2 2

Potential Energy ( )( )Force

mdxp mv mdt

mv dxT mdt

V xdV xF

dx

= =

=

= −2

. . . . - total energy = const

1 ( )2

K E P E E

dxm V x Edt

+ =

+ =

0

Initial conditions:

(0); t

dxxdt =

Initial conditions:(0); x E

Classical Trajectory Example:Particle in a gravitational field

x

0

x0

x0

V(x)

x0

E

Classical turning point

V(x) = -mgx

Potential Energy ( )( )Force

V x mgxdV xF mg

dx

=

= − = −

EOM (2nd Newton’s Law):2

2

d xm mgdt

= −

Solution: 20

1( )2

x t gt x= − +

( )2

202

Check:

1 2

d x d d dgt x gt gdt dt dt dt

= − + = − = −

Initial Conditions:Initial position

Initial velocity

0

0

(0)

0t

x xdxdt =

=

=

Classical Mechanics is a deterministic theory: once the ball is released, there is ZERO chance of finding it at any other positions than x(t) at time t.

K.E.

x

0

x0

t

x(t)

Classical Trajectory:Motion in a potential well

V(x) – PES (Potential Energy Surface)

EOM (Energy Conservation Law):

2

. . . . - total energy = const

1 ( )2

K E P E E

dxm V x Edt

+ =

+ =

Classically, there is ABSOLUTELY NO CHANCE to find the particle at any time beyond the classical turning points, i.e. at x<x1 or x>x2 .(This is referred to as the classically forbidden region).

x

V(x)

x1

E – total energy

Classical turning points

P.E.=V(x)K.E.

x2

The particle will oscillate back and forth between the classical turning points x1 and x2. The Potential Energy and Kinetic Energy will interconvert with the total energy staying constant.

x

x1

x2

t

Measuring electron charge: the oil drop experiment (1909)

Robert Millikan 1868 - 1953

The electric charge of any droplet is always a multiple of 1.6 x 10-19 Coulombs.

• The discovery of the electron lead to the first atomic model: J. J. Thomson proposed the “plum-pudding model”, where negatively

charged electrons swarm within a positively charged cloud. NO NUCLEUS

• 1910: Ernest Rutherford performed a landmark experiment that proved that the atom is mostly empty and its mass is concentrated in the nucleus.

1871 - 1937

• Rutherford’s Au-foil experiment results:– Most α-particles (+) went straight through.– But some (very few) α-particles were deflected.– They must have hit something relatively massive.

Atoms must be mostly empty, and they must contain an extremely dense and positive center (nucleus).

Atomic model based on Rutherford’s Au-foil experiment

• Nucleus is tiny – 10,000 times smaller than the atom • Atomic nucleus (+) balances the charges of the electrons (-).• Electrons are not part of the nucleus.• Nucleus is small but very heavy: a nucleus the size of a dot (.) would be as heavy as 2.5 Tons.• Materials around us are mostly empty. • Electron repulsions prevent us from passing through walls.

(-)

(+)

Classical Theory of Atom:Motion in a Coulomb potential1D: 3D:

2

0

1( )4

ZeV rrπε

= −

V(x)

E

V(x)

x

x0+ -

nucleuscharge +Ze

electroncharge -e

x +

-nucleuscharge +Ze

electroncharge -e

r

xy

z

V(r) r

E

The electron will oscillate back and forth between the classical turning points

The electron will orbit around the nucleus.

2

1,20

14

zexEπε

= ±

Note: - Total Energy E can assume any value under classical EOM

- Because oscillatory/orbital motion of e- involves acceleration and deceleration of an electric charge (e-), there will be radiation of EM waves. The atom will lose all of its energy through this emission and the electron will fall onto the nucleus! (Rutherford, 1911).

Historical Background of Quantum Theory

(1) Blackbody Radiation

2

2

8( ) Bk Tcπνρ ν =

0

( ) ???W dρ ν ν∞

∝ = ∞∫

Experimental observation:νmax∝ T(as the object gets hotter, it changes color from red to blue)Wavelength: λ=c/ν

Historical Background of Quantum Theory

(2) PhotoeffectTheory: Einstein (1905)

Historical Background of Quantum Theory(3) Compton effect (1923):

Scattering of light on particles (e.g., x-rays on electron)

The Fifth Solvay Conference(October 1927)

A. Piccard, E. Henriot, P. Ehrenfest, E. Herzen, Th. De Donder, E. Schrödinger, J.E. Verschaffelt, W. Pauli, W. Heisenberg, R.H. Fowler, L. Brillouin;P. Debye, M. Knudsen, W.L. Bragg, H.A. Kramers, P.A.M. Dirac, A.H. Compton, L. de Broglie, M. Born, N. Bohr;

I. Langmuir, M. Planck, M. Skłodowska-Curie, H.A. Lorentz, A. Einstein, P. Langevin, Ch. E. Guye, C.T.R. Wilson, O.W. RichardsonFifth conference participants, 1927. Institut International de Physique Solvay in Leopold Park.

29 participants, 17 Nobel Prize winners (one - twice)