PHYS-1050
1
Hydrogen Atom Energy Levels Solutions
Spring 2013
Introduction
Read through this information before proceeding on with the lab.
1.1
Energy Levels
1.1.1
Hydrogen Atom
A Hydrogen atom consists of a proton and an electron which are “bound” together—the proton (positive charge)
and electron (negative charge) stay together and continually interact with each other. If the electron escapes, the
Hydrogen atom (now a single proton) is positively ionized. Similarly, the Hydrogen atom can sometimes bind another
electron to it. Such a Hydrogen atom is negatively ionized. In astronomy, the former kind of ionization is much more
common.
In heavier atoms, the proton is replaced with a mixture of protons and neutrons collectively called the nucleus. The
nucleus of the Hydrogen atom is just one proton. Helium, on the other hand, has two protons and two neutrons for
a total of four nucleons (a “nucleon” is a general term for particles which are either a proton or neutron).
A Hydrogen atom is an electron and proton bound by the electromagnetic force (an attractive force between oppositely
charged particles). Because the electron is so much less massive than protons, early physicists visualized the electron
as being like a tiny planet which orbited the proton which acted like a tiny sun. Though this view has the advantage
of being easy to visualize, it is just an approximate physical representation.
A more physical view of the Hydrogen atom is one where the electron is not seen as orbiting the proton like a planet
around a sun, but exists as a diffuse cloud surrounding the nucleus. Only by measuring its location can one know
where the electron is (or rather was, as once the electron’s position is measured it moves to a different place). The
region where the electron is probably located is called the “electron cloud.” In some cases, the single most probable
electron-proton distance happens to correspond to the distance of the more planet-like model.
(a) Early and incorrect
view of the Hydrogen
atom.
(b) A slightly more accurate view of Hydrogen.
(c) Probability density of the electron for the ground and
two excited states of Hydrogen. Brighter regions are regions
where the electron is more likely to be found.
Figure 1: Models of the Hydrogen atom.
1.1.2
Orbitals
The density of this electron cloud at any location measures the probability of finding the electron there. In the
basic hydrogen atom, shown in Figure 1(c), the cloud is densest in the center and thins out with distance from the
nucleus, which means the electron is most likely to be found near the nucleus, in a region about 1/20 nm in size.
When additional energy is stored in the atom, the electron cloud takes on expanded patterns with low-density “nodal
surfaces” corresponding to the dark rings on the right two panels of Figure 1(c). We call these electron cloud patterns
“orbitals” (a term inherited from the early planet-like visualization) and each corresponds to a specific amount of
energy stored in the atom.
How much (kinetic) energy a planet has determines how far it is away from the star it orbits. Likewise, how much
energy an electron has determines how far it is way from the proton (or more accurately, how far out the electron
cloud extends). Planets can have essentially any energy and thus orbit at any distance. Electrons bound to the
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Hydrogen Atom Energy Levels Solutions
Spring 2013
nucleus, however, can not have just any value of energy. The electron can only occupy certain orbital “states,” each
with a specific amount of stored energy.
There is a lowest energy an electron can have and it corresponds to the state called the “ground state.” When the
electron (or atom) has higher energy than this lowest energy, it is said to be in an “excited state.”
An early model which mixed this discrete structure of electron energies states and the planet model is called the
Bohr model. Though the Bohr model doesn’t describe the electrons as clouds, it does a fairly good job of describing
the discrete energy levels. The Hydrogen Atom Simulator presented in this module is strongly based on the Bohr
Model.
1.1.3
Energy Levels
Because the states an electron occur only at discrete energy levels, they are said to be quantized. The word quantum
comes from a Latin word meaning “how much.” The branch of physics that provides the current model of the
Hydrogen atom is called quantum mechanics.
The electron in a Hydrogen atom can only have certain energies. These energies are called the Hydrogen’s “energy
levels.” The different energy levels of Hydrogen are denoted by the quantum number n where n varies from 1 for the
ground state (the lowest energy level) to ∞, corresponding to unbound electrons. In practice, electrons with high n
(e.g. 100 or more) are so weakly bound that even weak disturbances will pull the electron away.
Because it takes a minimum amount of energy, called the “ionization energy” to strip or ionize a bound electron
from the Hydrogen atom, energy levels are usually referred to as being negative quantities. In both classical physics
and quantum mechanics the absolute value of energy is irrelevant; only energy differences matter. It is convenient to
say that when ionized the electron will have zero binding energy to the proton. With this convention, the different
energy levels of a Hydrogen atom are given by the equation:
E=−
E0
n2
(1)
where E0 = 13.6 electron volts (eV) and n = 1, 2, 3 . . . and so on so that the ground state has energy E1 = −13.6
eV and the second energy level (the first excited state) has energy E2 = −13.6/4 eV = −3.4 eV. The equivalence
statements for electron volts, Joules (J), and calories (cal) are given by
1 eV = 1.602 × 10−19 J
1 J = 0.2390 cal
(2)
(3)
Note that the number of calories given in the Nutrition Facts on food is actually kilocalories, e.g. a snack bag of
Cool Ranch Doritos says it has 260 “calories,” which actually means it has 260 kilocalories, or
2.60 × 105 cal = 1.09 × 106 J = 6.79 × 1024 eV.
1.1.4
Excitation
A hydrogen atom with excess energy is said to be “excited.” The two primary ways to excite an atom are through
absorbing light and through collisions. When two atoms collide energy is exchanged. Sometimes, some of that
energy is used to excite an electron from a lower energy level to a higher energy level. How many collisions and how
energetic the collisions are will depend on how tightly the hydrogen atoms are spaced and their average temperature.
How absorbing light causes transitions is discussed more in Section 1.3. Another way to excite an atom is to absorb
electromagnetic energy, or in the terminology of quantum mechanics, “absorb a photon.”
1.2
1.2.1
Light
Properties of Light
We are all familiar with light. Through light we see the world. But what is it? Light can be thought of as “particles”
of electromagnetic energy called photons. Photons exhibit many properties that we are familiar with as particles.
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Hydrogen Atom Energy Levels Solutions
Spring 2013
Figure 2: Components of a Wave.
But light also exhibits wavelike properties (which property—wave or particle—it will manifest depends on how the
light is being observed). It is frequently convenient to express the properties of the light in wave terminology.
For light, the relationship between wavelength (λ) and frequency
(f ) is straightforward, their product is equal to the
speed of the wave, which for light is c ≈ 3.00 × 108 m/s . That is
c=λ×f
(4)
Another relationship that was instrumental in the formation of quantum mechanics was the relationship between the
energy of a photon at a certain frequency. It is
E =h×f
(5)
where h = 6.626 × 10−34 J·s is called Planck’s constant. Since the energy goes up as the frequency increases, the
energy is directly proportional to the frequency.
Because frequency and wavelength are related by a constant (c) the energy can also be written in terms of wavelength:
E = h × c/λ. When the energy increases the wavelength decreases and vice versa. That is, energy in inversely
proportional to wavelength.
In short, a photon can be described by either its energy, frequency, or wavelength. All three methods are frequently
used.
1.2.2
Electromagnetic Spectrum
Light can have a large range of values of energy/wavelength/frequency covering what is called the electromagnetic
spectrum. We often split the full spectrum into smaller regions and talk about the different “kinds” of light. Radio
waves are the least energetic kind of photons while gamma rays are the most energetic kind of photons. Figure
3 shows the electromagnetic spectrum. Note how small of a region the visible light we see with the human eye
covers.
Figure 3: The full spectrum of light.
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1.3
1.3.1
Hydrogen Atom Energy Levels Solutions
Spring 2013
Transitions
Absorbing Photons
According to the theory quantum mechanics, an electron bound to an atom can not have any value of energy, rather
it can only occupy certain states which correspond to certain energy levels. The formula defining the energy levels of
a Hydrogen atom are given by the Equation (1). The energy is expressed as a negative number because it takes that
much energy to unbind (ionize) the electron from the nucleus. It is common convention to say an unbound electron
has zero (binding) energy. Because an electron bound to an atom can only have certain energies the electron can
only absorb photons of certain energies exactly matched to the energy difference, or “quantum leap,” between two
energy states.
When an electron absorbs a photon it gains the energy of the photon. Because an electron bound to an atom can
only have certain energies the electron can only absorb photons of certain energies. For example an electron in the
ground state has an energy of −13.6 eV. The second energy level is −3.4 eV. Thus it would take
E2 − E1 = −3.4 eV − (−13.6 eV) = 10.2 eV
to excite the electron from the ground state to the first excited state.
If a photon has more energy than the binding energy of the electron then the photon will free the electron from the
atom—ionizing it. The ground state is the most bound state and therefore takes the most energy to ionize.
1.3.2
Emitting Photons
Generally speaking, the excited state is not the most stable state of an atom. An electron has a certain probability
to spontaneously drop from one excited state to a lower (i.e. more negative) energy level. When an electron drops
from a higher level to a lower level it sheds the excess energy, a positive amount, by emitting a photon in a random
direction.
1.3.3
Lyman, Balmer, Paschen Lines
Long before the Hydrogen atom was understood in terms of energy levels and transitions, astronomers had being
observing the photons that are emitted by Hydrogen (because stars are mostly Hydrogen). Atomic physicist Balmer
noted, empirically, a numerical relationship in the energies of photons emitted. This relationship was generalized and
given context by the Rydberg Formula. But the various discrete photon energies/wavelengths that were observed by
Balmer were named the Balmer series.
It was later understood that the Balmer lines are created by energy transitions in the Hydrogen atom. Specifically,
when a photon drops from an excited state to the second orbital, a Balmer line is observed. The Balmer series is
important because the photons emitted by this transition are in the visible regime. The Balmer series is indicated
by an H with a subscript α, β, γ, etc. with longest wavelength given by α.
As there are other transitions possible, there are other “series.” All transitions which drop to the first orbital (i.e.
the ground state) emit photons in the Lyman series. All transitions which drop to the 3rd orbital are known as the
Paschen series. Figure 4 shows some of the Lyman and Balmer transitions graphically.
2
Pre-Lab Questions
Complete the following questions before coming to class.
1. Complete Table 1 which relates the parameters of two different photons by circling the appropriate relationship.
The first row has been completed for you: “A red photon has a larger wavelength, smaller frequency, smaller
energy, and the same velocity through space as a blue photon.”
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Hydrogen Atom Energy Levels Solutions
Spring 2013
Figure 4: Energy Levels and Transition Lines.
Photon A
Red
Green
Infrared
Visual
X-Rays
Wavelength
Frequency
Energy
larger
the same
smaller
larger
the same
smaller
larger
the same
smaller
larger
the same
smaller
larger
the same
smaller
larger
the same
smaller
larger
the same
smaller
larger
the same
smaller
larger
the same
smaller
larger
the same
smaller
larger
the same
smaller
larger
the same
smaller
larger
the same
smaller
larger
the same
smaller
larger
the same
smaller
Velocity
(in space)
larger
the same
smaller
larger
the same
smaller
larger
the same
smaller
larger
the same
smaller
larger
the same
smaller
Photon B
Blue
Orange
Visual
Microwave
Gamma Rays
Table 1: Circle the appropriate relationships for each pair of photons.
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Hydrogen Atom Energy Levels Solutions
Spring 2013
2. Scientists often say “A is proportional to B” if B increases when A increases. They also say “A is inversely
proportional to C” if C decreases when A increases. Inspect the table above for evidence of such relationships
and use these terms to describe the relationships between wavelength, frequency, energy, and velocity below.
Solution: Wavelength is inversely proportional to frequency and energy. Frequency and energy are directly
proportional to each other. The velocity is constant regardless of any of the others.
3
Hydrogen Atom Simulator
Open the NAAP Hydrogen Atom Simulator from this link:
http://astro.unl.edu/naap/hydrogen/animations/hydrogen_atom.html
The Hydrogen Atom Simulator, shown in Figure 5, allows one to view the interaction of an idealized Hydrogen atom
with photons of various wavelengths. This atom is far from the influence of neighboring atoms and is not moving.
The simulator consists of four panels. Below gives a brief overview of the basics of the simulator.
Figure 5: NAAP Hydrogen Atom Simulator.
− The panel in the upper left shows the Bohr Model: the proton, electron, and the first six orbitals with the
correct relative spacing.
– The electron can absorb photons and jump higher energy levels where it will remain for a short time
before emitting a photon(s) and drop to lower energy level (with known probabilities fixed by quantum
mechanics).
– The electron can also be ionized. The simulator will a short time later absorb an electron.
– For convenience you can drag the electron between levels. Once it is released it will behave “physically”
once again as if it had gotten to that present level without being dragged.
− The upper right panel labeled “energy level diagram” shows the energy levels vertically with correct relative
spacing.
− The “Photon Selection” panel (bottom left) allows one to “shoot” photons at the Hydrogen atom. The slider
allows the user to pick a photon of a particular energy/wavelength/frequency.
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Hydrogen Atom Energy Levels Solutions
Spring 2013
– Note how energy and frequency are directly proportional and energy and wavelength are inversely proportional.
– On the slider are some of the energies which correspond to levels in the Lyman, Balmer, and Paschen
series.
– Clicking on the label will shoot a photon of that energy.
– If the photon is in visual band, its true color is shown. Photons of longer wavelengths are shown as red
and shorter wavelengths as violet.
− The “Event Log” in the lower right lists all the photons that the atom has encountered as well as all the electron
transitions.
– The log can be cleared by either using the button or manually dragging the electron to a particular energy
level.
For any particular level of the Hydrogen atom one can think of the photons that interact with it as being in three
groups:
Range 1:
None of the photons have enough
energy to affect the atom.
Increasing Energy −→
Range 2:
Some of the photons have the
right energy to make the electrons
jump to a higher energy level (i.e.
excite them).
Range 3:
All the photons have enough
energy to ionize the atom.
Note that the ranges are different for each energy level. Below is an example of the ranges for an electron in the
ground state of a Hydrogen atom.
Range 1:
0 eV to 10.1 eV
(10.2 eV needed to excite electron
to 1st orbital)
Ground State Electron of H
Range 2:
10.2 eV to 13.6 eV
(some will excite, some won’t)
Range 3:
> 13.6 eV
(anything greater than this will
ionize the electron)
When the simulator first loads, the electron is in the ground state and the slider is at 271 nm.
− Fire a 271 nm photon. This photon is in range 1.
− Gradually increase the slider to find a photon which is between range 1 and range 2 (for a ground state
electron). This should be the Lyman-α line (which is the energy difference between the ground state and the
second orbital).
− Increase the energy a bit more from the Lyman-α line and click “fire photon.” Note that nothing happens.
This is a range 2 photon but it doesn’t have the “right energy.”
− Increase the energy more until photons of range 3 are reached. In the simulator this will be just above the L
line.
– Technically there are photons which would excite to the 7th, 8th, 9th, etc. energy levels, but these are
very close together and those lines not shown on the simulator.
– The L line has an energy of −13.22 eV and is in range two. The ionization energy for an electron in the
ground state is 13.6 eV and so that is the correct range 3 boundary.
4
Analysis
1. Which photon energies will excite the Hydrogen atom when its electron is in the ground state? (Hint: there are
5 named on the simulator, though there are more.)
Solution: Lα = 10.2 eV, Lβ = 12.09 eV, Lγ = 12.75 eV, Lδ = 13.06 eV, L = 13.22 eV
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Hydrogen Atom Energy Levels Solutions
Spring 2013
2. Starting from the ground state, press the Lα button twice in succession (that is, press it a second time before
the electron decays). What happens to the electron?
Solution: The first photon causes the electron to jump from the ground state (1st level) to the first excited
state (2nd level). The second photon knocks the electron away; it ionizes the atom.
3. Complete the energy range values for the 1st excited state (i.e. the second orbital) of Hydrogen. Use the
simulator to fill out ranges 2 and range 3. The electron can be placed in the 1st orbital by manually dragging
the electron or firing an Lα photon once when the electron is in the ground state. Note also that the electron
will de-excite with time and so it may need to be placed in the 1st orbital repeatedly.
Range 1:
0 eV to 1.8 eV
(anything less than 1.9 eV
will fail to excite the atom)
1st Excited State (2nd Level) Electron of H
Range 2:
Range 3:
1.9 eV−3.4 eV
≥ 3.5 eV
4. What is the necessary condition for Balmer Line photons (Hα , Hβ , etc.) to be absorbed by the Hydrogen atom?
Solution: Per Figure 4, the electron must be at the 1st excited state (2nd level) for it to absorb a Balmer Line
photon.
5. Complete the energy range values for the 3rd orbital (2nd excited state) of Hydrogen. The electron can be placed
in the 3rd orbital by manually dragging the electron or firing an Lβ photon once when the electron is in the
ground state. Note also that the electron will de-excite with time and so it may need to be placed in the 3rd
orbital repeatedly.
Range 1:
< 0.65 eV
2nd Excited State (3rd Level) Electron of H
Range 2:
Range 3:
0.66 eV−1.5 eV
> 1.5 eV
(anything more than this will
ionize the atom)
6. Starting from the ground state, press two and only two buttons to achieve the 6th orbital in two different ways.
One of the ways has been given in Figure ??. Illustrate your transitions with arrows on the energy level diagram
in Figure ?? and label the arrow with the button pressed.
Solution: Lβ and Pγ photons will jump the electron from the ground (1st level) state to the 5th excited state
(6th level).
7. Press three buttons to bring the electron from the ground state to the 4th orbital. Illustrate the transitions as
arrows on the energy level diagram in Figure ?? and label the arrow with the button pressed.
Solution: Lα , Hα , and Pα photons will jump the electron from the ground (1st level) state to the 3rd excited
state (4th level).
8. How does the energy of a photon emitted when the electron moves from the 3rd orbital to the 2nd orbital compare
to the energy of a photon absorbed when the electron moves from the 2nd orbital to the 3rd orbital?
Solution: It’s exactly the same.
9. Find the amount of energy needed for the following 3 transitions.
− Lα : Level 1 to Level 2 −→ 10.20 eV
− Hα : Level 2 to Level 3 −→ 1.89 eV
− Pα : Level 3 to Level 4 −→ 0.66 eV
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Hydrogen Atom Energy Levels Solutions
Spring 2013
10. How much energy does a Lγ photon carry? What transition does the Lγ photon cause? How does the total
energy needed to undergo all three transitions in question 9 compare to the energy in a single Lγ photon?
Solution: Lγ = 12.75 eV, and it causes the electron to jump to the 3rd excited state (4th level). The sum of
the energies from all three transitions in question 9 equals the energy of one Lγ photon. So one Lγ photon can
cause the jump from 1st to 4th, or all three photons in this order —Lα , Hα , Pα —will cause the same transition
from 1st to 4th.
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