Lecture 16: Intro. to Quantum Mechanics
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Transcript of Lecture 16: Intro. to Quantum Mechanics
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Lecture 16: Intro. to Quantum Mechanics
• Reading: Zumdahl 12.5, 12.6
• Outline– Basic concepts.– A model system: particle in a box.– Other confining potentials.
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Quantum Concepts
• The Bohr model was capable of describing the discrete or “quantized” emission spectrum of H.
• But the failure of the model for multielectron systems combined with other issues (the ultraviolet catastrophe, workfunctions of metals, etc.) suggested that a new description of atomic matter was needed.
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Quantum Concepts• This new description was known as wave
mechanics or quantum mechanics.
• Recall, photons and electrons readily demonstrate wave-particle duality.
• The idea behind wave mechanics was that the existence of the electron in fixed energy levels could be though of as a “standing wave”.
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Exercise• What is the wavelength of an electron (mass 9.11
x 10-31 kg) traveling at a speed of 1.0 x107 m/s?
= h / p = h/mv
= 6.626x10-34 Js /(9.11x10-31kg)(1x107m/s)
= 6.626x10-34 Kgm2/s /(9.11x10-31kg)(1.x107m/s)
= 7.3x10-11 m
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Exercise• What is the wavelength of a baseball (mass 0.1 kg)
traveling at a speed of 35 m/s?
= h / p = h/mv
= 6.626x10-34 Js /(0.1kg)(35m/s)
= 6.626x10-34 Kgm2/s /(0.1kg)(35m/s)
= 1.9x10-34 m
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Uncertainty Principle• Another limitation of the Bohr model was that it
assumed we could know both the position and momentum of an electron exactly.
• Werner Heisenberg development of quantum mechanics leads him to the observation that there is a fundamental limit to how well one can know both the position and momentum of a particle.
x p h
4Uncertainty in position Uncertainty in momentum
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Example• Example:
What is the uncertainty in velocity for an electron in a 1Å radius orbital in which the positional uncertainty is 1% of the radius.
x = (1 Å)(0.01) = 1 x 10-12 m
p h
4x
6.626x10 34 J.s 4 1x10 12 m
5.27x10 23 kg.m /s
v pm
5.27x10 23 kg.m /s
9.11x10 31kg5.7x107 m
s huge
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Example• Example (you’re quantum as well):
What is the uncertainty in position for a 80 kg student walking across campus at 1.3 m/s with an uncertainty in velocity of 1%.
p = m v = (80kg)(0.013 m/s) = 1.04 kg.m/s
x h
4p
6.626x10 34 J.s 4 1.04kg.m /s
5.07x10 35 m
Very small……we know where you are.
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De Broglie’s wavelength• He provided a relationship between the electron properties and
their ‘wavelength’ which experimentally demonstrated by diffraction experiments
= h / p = h/mv
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Quantum Concepts (cont.)• What is a standing wave?
• A standing wave is a motion in which translation of the wave does not occur.
• In the guitar string analogy (illustrated), note that standing waves involve nodes in which no motion of the string occurs.
• Note also that integer and half- integer values of the wavelength correspond to standing waves.
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Quantum Concepts (cont.)• Louis de Broglie suggests that for the e- orbits envisioned by
Bohr, only certain orbits are allowed since they satisfy the standing wave condition.
not allowed
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Schrodinger Equation• Erwin Schrodinger develops a mathematical
formalism that incorporates the wave nature of matter:
ˆ H E
ˆ H The Hamiltonian:
ˆ p 2
2m (PE)
Kinetic Energy
The Wavefunction:x
E = energy
d2/dx2
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Wavefunction• What is a wavefunction?
= a probability amplitude
• Consider a wave:
y Ae i 2 t
y 2 Ae i 2 t Ae i 2 t A2Intensity =
• Probability of finding a particle in space:
*Probability = • With the wavefunction, we can describe spatial distributions.
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Potential Energy and Quantization• Consider a particle free to move in 1 dimension:
x
p “The Free Particle”
Potential E = 0
• The Schrodinger Eq. becomes:
ˆ H ˆ p 2
2m PE
(p)
ˆ p 2
2m(p)
p2
2m(p)
12
mv 2(p) E(p)
• Energy ranges from 0 to infinity….not quantized.
0
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Potentials and Quantization (cont.)• What if the position of the particle is constrained
by a potential:“Particle in a Box”
Potential E
x0
inf.
0 L = 0 for 0 ≤ x ≤ L
= all other x
• Now, position of particle is limited to the dimension of the box.
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Potentials and Quantization (cont.)• What do the wavefunctions look like?
x 2L
sin nxL
Like a standing wave
n = 1, 2, ….
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Potentials and Quantization (cont.)• What does the energy look like?
Energy is quantized
E
E n2h2
8mL2
n = 1, 2, …
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Potentials and Quantization (cont.)• Consider the following dye molecule, the length of which
can be considered the length of the “box” an electron is limited to:
E h2
8mL2 n final2 ninitial
2 h2
8m(8Å)2 22 1 2.8x10 19 J
What wavelength of light corresponds to E from n=1 to n=2?
L = 8 Å
700nm(should be 680 nm)
N
N
+
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Potentials and Quantization (cont.)• One effect of a “constraining potential” is that the
energy of the system becomes quantized.
• Back to the hydrogen atom:
e-
P+r
V (r) e2
r
constraining potential
r0
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Potentials and Quantization (cont.)• Also in the case of the hydrogen atom, energy becomes
quantized due to the presence of a constraining potential.
r0
Schrodinger Equation
V (r) e2
r
0
Recovers the “Bohr” behavior