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Objectives• Describe emission and absorption spectra and
understand their significance for atomic structure;• Explain the origin of atomic energy levels in terms
of the ‘electron in a box’ model;• Describe the hydrogen atom according to
Schrӧdinger;• Do calculations involving wavelengths of spectral
lines and energy level differences;• Outline the Heisenberg uncertainty principle in
terms of position-momentum and time-energy.
Atomic spectra• Hydrogen gas heated to high temps/exposed
to high electric field glows (EMITS light)⇒• Analyze light by
sending it through a spectrometer (splits light into component wavelengths)
• λ = 656 nm (Hα) red• λ = 486 nm (Hβ)
blue-green
Absorption Spectrum – the spectrum of light that has been transmitted through a gas.
White light (all wavelengths) passed
through hydrogen gas, then analyzed with a
spectrometer.The dark lines in the absorption spectrum are
at the exact same wavelengths as the
colored bright lines in the emission spectrum.
Atomic Spectra• Emission & absorption lines at specific
wavelengths for a particular gas• Scientific community: ?????• 1885 – Johann Balmer (accidentally) discovers
wavelengths in the emission spectrum of hydrogen given by ;– n is an integer (3,4,5…)– R is a constant– Scientific community STILL ?????
Atomic Spectra• Light carries energy ⇒ reasonable (based on
conservation of energy) to assume that the emitted energy is equal to the difference between the total energy of the atom before and after emission
• Emitted light consists of photons of a specific wavelength emitted energy must be a specific ⇒amount;
• Therefore, the energy of an atom is discrete (not continuous) but how could this be? How must our model change in light of this new evidence?
The ‘electron in a box’ model
Imagine that an electron is contained
within a “box” of linear size L. The
electron, treated as a wave, according to de
Broglie, has a wavelength associated
with it given by
Since the electron is confined to the box, it is
reasonable to assume that the electron wave is zero at
both edges of the box.
The ‘electron in a box’ model
In addition, since the electron cannot lose
energy, it is also reasonable to assume
that the wave associated with the
electron in this case is a standing wave.
So we want a standing wave that will have nodes at x = 0 and x = L. This implies that
the wavelength must be related to the size L of the
box through
The ‘electron in a box’ model
Therefore
This result shows that, because we treated the electron as a standing wave in a ‘box’, we deduce that the electron’s energy is ‘quantized’ or discrete, i.e. it
cannot have any arbitrary value.
The ‘electron in a box’ modelThe electron’s Ek can only be
• This model gives us a discrete set of energies• Not a realistic model for an electron in an atom, but it
does show the discrete nature of the electron energy when the electron is treated as a wave; points toward
the correct answer.
The Schrӧdinger Theory• 1926 – Austrian Physicist Erwin Schrӧdinger• Assumes as a basic principle that there is a wave
associated to the electron (like de Broglie), called the wavefunction, ψ(x,t).
• The wavefunction is a function of position x and time t.
• Given the force that act on an electron, it is possible, in principle, to solve a complicated differential equation obeyed by the wavefunction (the Schrӧdinger equation) and obtain ψ(x,t).
The Schrӧdinger Theory• For example, there is one wavefunction for a free
electron, another for an electron in the hydrogen atom, etc.The interpretation of what ψ(x,t) really
means came from German physicist Max Born. He suggested that |ψ( , )|𝒙 𝒕 𝟐 (the
square of the absolute value of ψ(x,t) can be used to find the probability that an
electron will be found near position x at time t.
The Schrӧdinger Theory• The theory only gives probabilities for finding an
electron somewhere – it does not pinpoint and electron at a particular point in space; a radical change from ordinary (classical) physics where objects have well defined positions.
• When the Schrӧdinger theory is applied to the electron in a hydrogen atom it gives results similar to the simple electron in a box example of the previous section.
The Schrӧdinger Theory• It predicts that the total energy of the electron is
given by ; where n is an integer that represents the energy level the electron inhabits and C is a constant equal to ; k is the constant in Coulomb’s law, m is the mass of the electron, e is the charge of the electron and h is Planck’s constant.
• ; theory predicts that the electron in the hydrogen atom has quantized energy
The Schrӧdinger Theory
• High n energy levels are very close together
• When an electron absorbs a photon, it jumps up an energy level (absorption spectra)
• When the electron loses enough energy to drop down one or more levels, it emits a photon of energy equal to the energy it lost (emission spectra)
ExampleShow how the formula for the electron energy in the Schrӧdinger theory can be used to derive the
empirical Balmer formula mentioned earlier .
The Schrӧdinger TheoryThe variation of the probability distribultion function (pdf) with distance r from the nucleus for the n=1 (lowest) energy level of the hydrogen atom. The height of the graph is proportional to |ψ( , )|𝒙 𝒕 𝟐.The shaded area is the probability
for finding the electron at a
distance from the nucleus between
r=a and r=b.
The Heisenberg Uncertainty Principle• Discovered 1927 - Named after Werner
Heisenberg (1901 – 1976); one of the founders of quantum mechanics
• Founding idea: wave-particle duality – particles sometimes behave like waves and waves sometimes behave like particles, so that we cannot cleanly divide physical objects as either particles or waves
The Heisenberg Uncertainty Principle• The Heisenberg uncertainty principle applied to
position and momentum states that it is not possible to measure simultaneously the position and momentum of something with indefinite precision – representative of a fundamental property of nature; nothing to do with equipment.
• ; Δx – uncertainty in position, Δp – uncertainty in momentum, h = planck’s constant
• Making momentum as accurate as possible makes position inaccurate. If one is zero, the other is infinite.
The Heisenberg Uncertainty Principle• Imagine: electrons emitted from a hot wire in a
cathode ray tube (crt) and we try to make them move in a horizontal straight line by inserting a metal with a small opening of size a. we can make the electron beam as thin as possible by making the opening as small as possible – electrons must be somewhere within the opening so Δx < a.
• a should not be on the same order of magnitude as the de Broglie wavelength of the electrons to avoid diffraction.
The Heisenberg Uncertainty Principle• Here, too, the electron will diffract through the
opening some electrons emerge in a ⇒direction that is no longer horizontal.
• We can describe this phenomenon by saying that there is an uncertainty in the electron’s momentum in the vertical direction of magnitude Δp
The Heisenberg Uncertainty Principle• The angle by which the electron is diffracted is
given by a = opening size = uncertainty in position = Δx. From the figure
The Heisenberg Uncertainty PrincipleApplication: consider an electron which is known to
be confined within a region of size L. Then the uncertainty in position must satisfy Δx < L, so Δp must be and . Applying this to an electron in the
hydrogen atom (L≈10-10m): Which is the correct order of magnitude value of
the electron’s kinetic energy.
The Heisenberg Uncertainty Principle
Note the resemblance of this formula () to the formula for the energy obtained earlier in the ‘electron in a box’ model (). Apart from a few
numerical factors (of order 1) the two are the same, indicating the basic connection between the
uncertainty principle and duality.
The Heisenberg Uncertainty Principle• Also applicable to energy and time.• The Heisenberg uncertainty principle
applied to energy and time states that it is not possible to know simultaneously the energy and time of something with indefinite precision
• ; ΔE – uncertainty in energy, Δt – uncertainty in time, h = planck’s constant