Download Chapter 28 Atomic Physics Wave Function, ψ The Heisenberg

Wave Function,
Function, ψ
Chapter 28 Atomic Physics
The value of Ψ2 for a particular object at a certain place
and time is proportional to the probability of finding the
object at that place at that time.
The Hydrogen Atom
The Bohr Model
Electron Waves in the Atom
The Heisenberg Uncertainty Principle
Momentum and position DxDp ≥ h/4p
Energy and time DEDt ≥ h/4p
Spectrum of White Light
Emission Spectrum of Hydrogen
Emission Spectrum of Helium
Emission Spectrum of Lithium
Emission Spectrum of Mercury
Absorption Spectrum of Hydrogen
Line Spectrum
Rutherford Model Predicts:
(1)A continuous range of frequencies of light emitted
(2)Unstable atoms
These are inconsistent with experimental observations
Why ?
Quantized orbits
Each orbit has a different energy
Niels Henrik David Bohr
1885-1962
Excited Electron
Photon emitted: hf=Eu-El
1/λ=R(1/22-1/n2), n=3,4,… for Balmer series
where Rydberg constant R=1.097x107 m-1
Equations Associated with The Bohr Model
Spectrum of White Light
Electron’s angular momentum
L=Iω=mvrn=nh/2π, n=1,2,3
n is called quantum number of the orbit
Emission Spectrum of Hydrogen
Emission Spectrum of Helium
Emission Spectrum of Lithium
Emission Spectrum of Mercury
Absorption Spectrum of Hydrogen
Line Spectrum: hf=Eu-El
Radius of a circular orbit
rn=n2h2/4π2mkZe2=(n2/Z)r1
where r1=h2/4π2mke2=5.29x10-11 m (n=1)
r1 is called Bohr radius, the smallest orbit in H
Total energy for an electron in the nth orbit:
En=(-2π2Z2e4mk2/h2)(1/n2)=(Z2/n2)E1
where E1=-2π2Z2e4mk2/h2 =-13.6 eV (n=1)
E1 is called Ground State of the hydrogen
Both orbits and energies depend on n, the quantum number
To break a hydrogen atom apart requires 13.6 eV
+ 13.6 eV =
Hydrogen
atom
+
Proton
e
electron
Question: In the Bohr model of the hydrogen atom,
the electron revolves around the nucleus in
order to
(a) emit spectral lines
(b) produce X rays
(c) form energy levels that depend on its speed
(d) keep from falling into the nucleus
v=2.2x106 m/s
p
Electron orbit
r=0.053 nm
Question: A hydrogen atom is in its ground state
when its orbital electron
(a) is within the nucleus
(b) has escaped from the atom
(c) is in its lowest energy level
(d) is stationary
Answer: d
X
Which one of the following statements is the assumption
that Niels Bohr made about the angular momentum of
the electron in the hydrogen atom?
(a) The angular momentum of the electron is zero.
(b) The angular momentum can assume only certain
discrete values.
(c) Angular momentum is not quantized.
(d) The angular momentum can assume any value
greater than zero because it’s proportional to
the radius of the orbit.
(e) The angular momentum is independent of the mass
of the electron.
Answer: c
Example: Find the orbital radius and energy
of an electron in a hydrogen atom
characterized by principal quantum
number n=2.
Solution: For n=2,
r2=r1n2=0.0529nm(2)2=0.212 nm
and
E2=E1/n2=-13.6/22 eV=-3.40 eV
X
X
1.The kinetic energy of the ground state electron in
hydrogen is +13.6 eV. What is its potential energy?
(a) –13.6 eV
(b) –27.2 eV
(c) zero eV
(d) +27.2 eV
(e) +56.2 eV
2. An electron is in the ground state of a hydrogen atom. A
photon is absorbed by the atom and the electron is excited
to the n = 2 state. What is the energy in eV of the photon?
(a) 13.6 eV
(b) 3.40 eV
(c) 0.54 eV
(d) 10.2 eV
(e) 1.51 eV
Line and Absorption Spectra
hf=Eu- El
hc/λ=Eu - El
1/λ=(1/hc)Eu- El
1/λ=(2π2Z2e4mk2/ch3)(1/nl2-1/nu2)
Each atom in the periodic table has a unique set of
spectral lines. Which one of the following
statements is the best explanation for this
observation?
(a) Each atom has a dense central nucleus.
(b) The electrons in atoms orbit the nucleus.
X (c) Each atom has a unique set of energy levels.
(d) The electrons in atoms are in constant motion.
(e) Each atom is composed of positive and negative
charges.
Complete the following statement: An individual copper
atom emits electromagnetic radiation
with wavelengths that are
(a) evenly spaced across the spectrum.
(b) unique to that particular copper atom.
(c) the same as other elements in the same column of the
periodic table.
X (d) unique to all copper atoms.
(e) the same as those of all elements.
WaveWave-Particle Duality
Orbits and energies are quantized!
λ = h/p= h/mv
Why?
Condition for orbit
The quantized orbits and energy states in
the Bohr model are due to the wave nature
of the electron, and the electron wave
functions can only occur in the form of
standing waves.
An electron can circle an atomic nucleus
only if its orbit is a whole number of
electron wavelengths in circumference
Implication: The wave-particle duality is at
the root of atomic structure
Condition for orbit stability
nλ=2πrn, n=1,2,3…
Unless a whole number of wavelengths fits into the wire loop,
destructive interference causes the vibrations to die out rapidly
de Broglie wavelength is λ=h/mv and the speed of the electron in a
hydrogen is
v=2.2x106 m/s
so λ=h/mv
=6.63x10-34Js/(9.1x10-31kg)(2.2x106m/s)
=3.3x10-10 m
2πr1=2πx5.29x10-11m=3.3x10-10 m
The orbit of the electron in a hydrogen atom corresponds to
one complete electron wave joined on itself!
n=4
Question: With increasing quantum number, the
energy difference between adjacent energy
levels
(a) decreases
(b) remains the same
(c) increases
(d) sometimes decreases and sometimes
increases
Answer: a
Question: The bright-line spectrum produced by
the excited atoms of an element contains
wavelength that
(a) are the same for all elements
(b) are characteristic of the particular element
(c) are evenly distributed throughout the entire
visible spectrum
(d) are different from the wavelength in its darkline spectrum
Answer: b
Question: How can the spectrum of hydrogen
contains so many lines when hydrogen contains
only one electron?
Answer: The electron in the hydrogen atom can
be in any of a nearly infinite number of quantized
energy levels. A spectral line is emitted when the
electron makes a transition from one discrete
energy level to another discrete energy of lower
energy. A collection of many hydrogen atoms
with electrons in different energy levels will give
a large number of spectral lines.
Question: An atom emits a photon when one of its
electrons
(a) collides with another of its electrons
(b) is removed from the atom
(c) undergoes a transition to a quantum state of
lower energy
(d) undergoes a transition to a quantum state of
higher energy
Answer: c
Question: According to the Bohr model, an
electron can revolve around the nucleus of a
hydrogen indefinitely if its orbit is
(a) a perfect circle
(b) sufficient far from the nucleus to avoid capture
(c) less than one de Broglie wavelength in
circumference
(d) exactly one de Broglie wavelength in
circumference
Answer: d
Example: An electron collides with a hydrogen
atom initially in its ground state and excites it to
a state of n=3, How much energy was
transferred to the hydrogen atom in this inelastic
(KE not conserved) collision?
Solution: ΔE=E1(1/n2f-1/n2i)
Here ni=1, nf=3 and E1=13.6 eV
ΔE=E1(1/n2f-1/n2i)=E1(1/32-1/12)=-13.6(-8/9)eV
=12.1 eV
Early Quantum Theory
¾ The presence of definite energy levels in an atom is true
for all atoms. Quantization is characteristic of many
quantities in nature
¾ Bohr’s theory worked well for hydrogen and for oneelectron ions. But it did not prove as successful for
multielectrons.
¾ It is quantum mechanics that finally solved the problems
¾ Quantum energy: E=hf
¾ Photoelectric effect: hf=KEmax+Wo
¾ De Broglie wavelength: λ=h/mv
¾ Bohr theory: L=mvr=nh/2π
En=E1/n2 where E1=-13.6 eV
¾ Wave-particle duality
Limitations of the Bohr Theory
¾Unable to predict the line spectra for more
complex atoms
¾Unable to predict the brightness of
spectral lines of hydrogen
¾Unable to explain the fine structure
¾Unable to explain bonding of atoms in
molecules, solids and liquids
¾Unable to really resolve the wave-particle
duality
Bohr Model
Quantum Mechanics
¾It solves all these problems and has
explained a wide range of physical
phenomena.
¾It works on all scales of size. Classical
physics is an approximation of quantum
physics
¾It uses an abstract mathematical
formulation dealing with probabilities
Quantum mechanics
Upon which one of the following parameters does the energy of a photon
depend?
(a) mass
(c) polarization
X
(b) amplitude (d) frequency
(e) phase relationships
For which one of the following problems did Max Planck make
contributions that eventually led
to the development of the “quantum” hypothesis?
Definite circular orbits of electrons
No precise orbits of electrons, only the
probability of finding a given electron
at a given point
(a) photoelectric effect
(d) the motion of the earth in the ether
(b) uncertainty principle
vacuum
(e) the invariance of the speed of light through
X
(c) blackbody radiation curves
Description of waves
Wave Function, ψ
The Heisenberg Uncertainty Principle
Type of waves
Water waves
Sound waves
Light waves
Matter waves
Variable physical quantity
Height of the water surface
Pressure in the medium
Electric and magnetic fields
Wave function, Ψ
Ψ, the amplitude of a matter wave, is a
function of time and position
Probability Density Ψ2
Why Ψ2? Why not Ψ?
The value of Ψ2 for a particular object at a certain place
and time is proportional to the probability of finding the
object at that place at that time.
¾Amplitude of every wave varies from –A to
+A to –A to +A and so on (A is the
maximum absolute value whatever the
wave variable is).
¾ A negative probability is meaningless.
¾ Ψ2 gives a positive quantity that can be
compared with experiments.
For example:
Ψ2 =1: the object is definitely there
Ψ2 =0: the object is definitely not there
Ψ2 =0.4: there is 40% chance of finding the object there
at that time.
Ψ2 starts from Schrodinger’s equation, a differential
equation that is central to quantum mechanics
The key point to the wave function is that the position
of a particle is only expressed as a likelihood or probability
until a measurement is made.
The probability the electron will be found at the particular position
is determined by the wave function illustrated to the right of the aperture.
When the electron is detected at A, the wave function instantaneously
collapses so that it is zero at B.
Example: Compare the de Broglie wavelength of 54-eV electrons with
that of a 1500-kg car whose speed is 30 m/s.
The Heisenberg Uncertainty Principle
Solution:
For the 5454-eV electron:
electron
KE=(54eV)(1.6x10-19J/eV)=8.6x10-18 J
2
KE=1/2 mv , mv=(2mKE)1/2
λ=h/mv=h/(2mKE)1/2= 1.7x10-10 m
The wavelength of the electron is comparable to atomic scales (e.g.,
Bohr radius=5.29x10-11 m). The wave aspects of matter are very
significant.
For the car:
λ=h/mv=6.63x10-34 J•s/(1.5x103)(30m/s)= 1.5x10-38 m
The wavelength is so small compared to the car’s dimension that no
wave behavior is to be expected.
The Heisenberg Uncertainty Principle
In the microscopic world where the wave
aspects of matter are very significant,
these wave aspects set a fundamental
limit to the accuracy of measurements
of position and momentum regardless of
how good instruments used are.
The uncertainty principle is the physical
law which follows from the wave nature
of matter
1. If an object has a well-defined position at
a certain time, its momentum must have
a large uncertainty.
2. If an object has a well-defined
momentum at a certain time, its position
must have a large uncertainty.
p=h/λ precise
x unknown
Δx better defined
(narrower wave packet)
Δp less defined
(greater spread of λ)
Uncertainty principle ΔxΔp≥h/2π
Uncertainty Principle
Momentum and position ΔxΔp ≥ h/2π
Energy and time ΔEΔt ≥ h/2π
Question: The quantum theory of the atom
(a) is based on the Bohr theory
(b) is more comprehensive but less accurate
than Bohr theory
(c) cannot be reconciled with Newton’s laws
of motion
(d) is not based on a mechanical model and
considers only observable quantities
Answer: d
Question: A moving body is described by the
wave function Ψ at a certain time and
place. The value of Ψ2 is proportional to
the body’s
a. electric field.
b. speed
c. energy
d. probability of being found
Answer: d
Question: Modern physical theories indicate
that
a. all particle exhibit wave behavior
b. only moving particles exhibit wave
behavior
c. only charged particles exhibit wave
behavior
d. only uncharged particles exhibit wave
behavior
Answer: b
Question: A large value of the probability
density Ψ2 of an atomic electron at a
certain place and time signifies that the
electron
(a) is likely to be found there
(b) is certain to be found there
(c) has a great deal of energy there
(d) has a great deal of charge there
Answer: a
Question: The narrower the wave packet of
a particle is
a. the shorter its wavelength
b. the more precisely its position can be
established
c. the more precisely its momentum can be
established
d. the more precisely its energy can be
established
Answer: b
Question:The description of a moving body
in terms of matter wave is legitimate
because
a. it is based on common sense
b. matter waves have been actually seen
c. the analogy with EM waves is plausible
d. theory and experiment agree
Answer: d
Question: The wave packet that
corresponds to a moving particle
a. has the same size as the particle
b. has the same speed as the particle
c. has the speed of light
d. consists of x-ray
Answer: b
Question: If Planck’s constant were larger than it
is,
a. moving bodies would have shorter wavelength
b. moving bodies would have higher energies
c. moving bodies would have higher momenta
d. The uncertainty principle would be significant on
a larger scale of size
Answer: d
Wave Function, ψ
The Heisenberg Uncertainty Principle