Download De Broglie Waves.

De Broglie Waves.
 Bohr's success in describing the line spectrum of hydrogen atom
when he assumed that the electron moves in orbit (circular shape)
around stationary nucleus by angular momentum (quantized values)
 In classical physics, the wave length of
standing wave are quantized, that for circle (ring)
1
 From Einstein relations
o Photoelectric effect
o Theory of relativity
So
Photons
Particles
 De Broglie suggests that moving particles accompanied by a wave
with wave length can be calculated from above equation. He also
assumed that this relation be hold for any motion of any particle
not only for electrons. That is called the matter waves or the dual
nature
.
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Matter (e.g. electrons)
Classical: matter behaves as particles
Quantum: matter particles also behave like waves
Macroscopic particles
Large mass
large momentum
wavelength extremely small
wave-like properties cannot be observed
 Boher Assumption
From equations 1 and 2
And so
 Examples
 Calculate the de Broglie wavelength for a baseball 0.14 kg
traveling at 40 m/s.
 What is the de Broglie wavelength (in nm) associated with a 3.50
g Ping-Pong ball traveling 56.0 km/h?
 Calculate the de Broglie wavelength of an electron traveling at
1% of the speed of light. (me= 9.11 x 10-31kg)
Schrödinger Equation
1. The time independent Classical wave equation
2. De Broglie Relation
3. Total Energy
From 1 and 2
From 3
(
)
So
(
)
(
(
)
)
̂
Time
Independent
Schrodinger
Equation
 Schrödinger Equation in 3 Dimensions
 In 3-D: wave function varies in 3 dimensions defined by
Cartesian coordinates (x, y, z)
 Hamiltonian operator:
̂
(
̂(
)
)
 Laplace operator (Laplacian) “del squared”:
(
)
 Schrödinger Equation in 3 Dimensions
̂
̂
 Solution of Schrodinger equation (SWE)
 The solution of the equation means that we must know the
Hamiltonian operator to get the wave function and total energy
 The wave functions contain all the dynamical information about
systems they describe (position, momentum, angular momentum,
energy)
 The wave functions must be well-behaved
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Restriction on acceptable wave functions
only certain solutions to SWE are acceptable
only certain corresponding energies are possible
Quantization
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A wave function (ψ) doesn’t have a direct physical meaning
BUT the square of the wave function does!
Max Born deduced that the probability of finding electron is ψ2
Probability of finding particle in a certain volume of space
To calculate probability →integral in 3-D
∫