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8/2/2019 Ch 7-3 Simple QM
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Quantum Mechanics
Where did it come form?
It may look like pulling a rabbit out of a hat
. . .but it is not.
8/2/2019 Ch 7-3 Simple QM
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Quantum mechanics was developed to understand the
properties of atoms and molecules.
In quantum mechanics physicists use the wave propertiesof matter to understand atoms.
The main equation is called the Schrödinger equation
• Scientists understood waves ( ).
• Scientists understood particles could behave like
waves ( λ = h/p).
• Schrödinger combined these two equations to
develop the Schrödinger equation.
22
2
2( )
d y y x
dx
π
λ
⎛ ⎞= − ⎜ ⎟
⎝ ⎠2
( ) sin( ) x
y x Aπ
λ
=
To understand QM, you need to understand standing waves.
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STANDING WAVES
Plucking a guitar string is an example of a standing wave.
Because the ends of the string are fixed, only certain
wavelengths are allowed.
The longest wavelength occurs when the string is plucked in
the middle. In this case the wavelength is twice the length of
the string.
This is illustrated in the following slides.
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Standing wave fixed at 0 and 10; maximum amplitude is 0.2.
Wavelength = λ = 2L = 20
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0 2 4 6 8 10
Length
A
m p l i t u d e
(click through these slides at high speed for a simulation of a plucked stretched string.)
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-0.2
-0.15
-0.1
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0
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0.15
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-0.2
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OTHER POSSIBLE STANDING WAVES
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OTHER POSSIBLE STANDING WAVES
2
3
Lλ =
2
2
L Lλ = =n = 2
___
An = 3
Nodes
L
2 21
L Lλ = =n = 1
0
The greater the number of nodesthe greater the frequency
Reger
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Functions that are solutions to the wave equation are
given the Greek letter Ψ (psi).
Classically, |ψ|2 α intensity for a wave
Quantum mechanically,
|ψ|2 = probability particle at that location
2
1dxΨ =∫
Example: A stretched string of length L (like a guitar string) is
analogous to a particle (say an electron) constrained to travel back
and forth along a line of length L (called a 1 dimensional box).
Results for 1 dimensional particle in a box
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p
ψE |ψ|2
Quantum
Mechanics
Classicalany E
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Some observations or thoughts:
The results of the Schrödinger equation are not part of our
everyday experience and not at all obvious or easy to
understand.
Its usefulness is determined by testing it on atoms and
molecules.
It turns out to work well and is an extremely useful model.
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