5.1 Introduction: Spectral Lines and the Quantum Staircase
The research questions for this lab are both technical and conceptual:
Calibration: How well calibrated are the USB 400 Ocean Optics Spectrometers?
Quantum Connection: How are observed spectral lines related to Energy Level Diagrams (ELDs)?
These questions refer to the quantum theory of spectra (Q11 in Moore's text) in atoms and molecules, a topic typically not yet reached at this time of the semester. A very brief introduction to `spectral lines' and energy level diagrams is in order before introducing the experimental tasks that will help us address both research questions.
A spectrum serves as a unique fingerprint of the chemical identity of an atom, that is, its elemental identy, whether Hydrogen, Helium, Lithium, etc.. The underlying physics is that an atom has a discrete, unique set of allowed energy eigenstates (or levels). When an electron transitions (or "jumps") between two allowed states, the energy difference is emitted (or absorbed) as a single photon. In even the most complex instances, the quantum jumps occur between two energy levels or states. A simple way to picture the idea is shown below, something one might see in a science museum exhibit (I LOVE SCIENCE MUSEUMS). Although the language is French, it's not hard to see what is going on.
Figure 1. An exhibit displaying spectral lines in Hydrogen, both in emission and absorption. The absorption spectrum is shown as an measured, sharp, omission of light used to excite atoms prepared to be in a relatively low energy state. This science museum must've been in France!
We have until now conceived of a silver atom undergoing an $SGA_z$ measurement as solely determining the spin angular momentum of an electron in a silver atom, the `outermost' one. But also, by virtue of the strength of the magnetic field in the Stern Gerlach Analyzer (as opposed to its gradient), there is also an energy difference between the spin up and spin down states, because $U = - \vec{\mu} \cdot \vec{B}$, and so our silver atom in a Stern-Gerlach Analyzer may be characterized as having 2 discrete energy levels between which quantum jumps can happen at frequencies that satisfy what will later be called the Bohr frequency, $\omega_0 = (E_+ - E_-)/\hbar$. And this reminds of Planck's formula for the energy of photons.
The energy of the emitted photon ($E_{ph}$) corresponds precisely to the energy difference between two states (for the most general circumstance, we'll label these as $E_i$ and $E_k$):
$$ E_{ph} = \hbar \omega_{ik} = \hbar \cdot c \cdot {\cal k}_{ik} = \frac{hc}{\lambda_{ik}} \equiv E_i - E_k.$$
The wavelength $\lambda_{ik}$ (the observed spectral line) is determined by the energy difference between two states, not the energy of a single state. This is the profound, unclassical idea we will explore. An historical discussion of this unclassical behavior is taken up at the end, but now we must turn to the experimental tasks themselves.
Safety and Setup Precautions for the Ocean Optics Spectrometer and Gas Discharge tubes
UV Radiation: The lamps emit considerable ultraviolet radiation. Do not stare at the bulbs.
High Voltage: The potential is about 5000 V. Do not touch the discharge apparatus when the power is on.
Detector Saturation: The fiber optic cable can saturate the detector. Consult your instructor to adjust the integration time or use the attenuation filter controls.
Labeling:Record the element's name from the label on the tube itself, not the power supply housing.
Figure 1. On the left, a look into our modular spectrometer, showing its principal parts, and on the right, the basic setup with the plasma discharge lamp, fiber optics for light collection, and so forth.
5.2 Task 1: Calibration and Resolution
First, an example of a spectrum acquired of the fluorecent lamps that light the building we are in! The figure below can be found in the Wikipedia article on the fluorescent lamp (scroll down to see the spectra). There you'll see
Figure 2. Typical fluorescent lamp spectra, and table of 'spectral lines' and wavelengths: note the figure caption indicating that the spectra were taken using the Ocean Optics USB2000. My apologies for not reproducing the references from the Wikipedia article.
They mention a calibration discrepancy! That's our first research question. What is the calibration discrepancy for ours?
Data Acquisition (H, He, and Hg):
Obtain a spectrum for atomic Hydrogen (H), Mercury (Hg), and Helium (He) in the 400 nm – 900 nm range.
Annotation: Label the 4 strongest lines visible in each spectrum with a lowercase letter (e.g., a, b, c, d). Print the annotated plots and affix them to your lab notebook.
``Know-How'' Capture: Record the exact sequence of software commands, settings, and file path names used to acquire and save the data. We are getting used to a new instrument for this lab (which we'll use for the next two labs), and we need to capture the procedures we'll use! For a screenshot of the OceanView Software GUI with some helpful notes, look
here . Your instructor will have some instructions too.
Calibration Table:
For the 4 strongest lines of each element, create a table using the NIST Atomic Spectra Database for comparison.
Table Columns: Include Measured Wavelength ($\lambda_{\text{exp}}$), $\lambda_{\text{exp}}$ Uncertainty ($\delta\lambda_{\text{exp}}$), NIST Wavelength ($\lambda_{\text{NIST}}$), and Discrepancy ($\Delta\lambda = \lambda_{\text{exp}} - \lambda_{\text{NIST}}$).
Uncertainty: Explain in detail how you estimated the uncertainty $\delta\lambda_{\text{exp}}$ for your measured peak wavelengths.
Analysis: Based on your data, quantitatively assess the spectrometer's **calibration**. Is there evidence of a systematic uncertainty? How does the discrepancy compare to your measured uncertainty? Your interpretation must be summarized in your abstract.
5.3 Task 2: Spectral Lines and Energy Level Diagrams
Create Partial ELDs:
Qualitative Sketch:On your own, in your lab notebook, create a **qualitatively correct sketch** of a partial ELD (Grotrian diagram) for one of your elements that **accounts only for the 4 strongest lines** you measured. Your vertical gaps should visually represent the relative energy differences. and **annotate your ELD** by drawing arrows to show the 4 transitions that correspond to your 4 measured spectral lines. For this lab we won't worry about angular momentum states!
Verification:Use the NIST energy levels tool to find the corresponding energy levels for your chosen lines. Get an image of, or redraw it in your lab notebook nearby your own sketc, incorporating the accurate relative level positions. Note: The NIST diagram calculates the energy gaps of the chosen lines (your instructor will demonstrate this). Are these consistent with your acquired spectrum?
Conceptual Questions:
Minimum Levels: What is the minimum number of energy levels required to account for $N=4$ spectral lines? State your assumption.
Internal Consistency: How can you verify that your drawn ELD is consistent with your measured spectra? (Hint: Relate the relative gaps on your sketch to the numerical values of the measured wavelengths.)
Plasma Excitation: In the discharge tube, how are the atoms excited (what is being accelerated, and how is energy delivered)?
Individual Atom: Does an individual atom emit the entire spectrum of the element? Explain in terms of the quantum processes (jumps) that create the spectrum.
Absorption vs. Emission: Would the absorption spectrum of this element have the same lines as the observed emission spectrum? Explain the physical difference between the two types of spectra and briefly explain how one would measure the absorption spectrum.
A final note about the abstract. To quote from the lab notebooks rubric, ``An abstract...''
quotes principal results, that permits
comparisons of uncertainties and discrepancies (if that is possible), using significant figures appropriately, and
describes the essence of the methods (in not more than a sentence or two!) used to obtain the results for those who are not enrolled in the laboratory course–you can assume they are students in the lecture course though, and
states (as simply and directly as possible) an answer to the research questions for each task of the
experiment that is supported by the experimental work presented.
For task 1, the discrepancies are clearly the featured result, while for task 2, it's the verification part, comparing qualitatively the sketches you drew of the ELD's and the NIST generated ones. The NIST diagrams also permit a comparison between the wavelength observed and the NIST calculated wavelength between the two levels corresponding to your 4 chosen strongest lines.
5.4 Historical Background
Our research questions are quite practical and theoretical, having to do with the quantum theory of radiation in atoms and molecules, and an imaginary conceptual construct known as an Energy level diagram, or rather a Grotrian Diagram [1].
Since we have not yet reached the chapters in Moore's text treating spectra in atoms and molecules (but do please look at Q11 in Moore's text!), a quote
from Dirac's monograph on Quantum Mechanics [2] helps us understand how understanding spectral lines required a marked departure from classical physics:
The necessity for a departure from classical mechanics is clearly shown by experimental results. In the first place the forces known in classical electrodynamics are inadequate for the explanation of the remarkable stability of atoms and molecules, which is necessary in order that materials may have any definite physical and chemical properties at all. The introduction of new hypothetical forces will not save the situation, since there exist general principles of classical mechanics, holding for all kinds of forces, leading to results in direct disagreement with observation. For example, if an atomic system has its equilibrium disturbed in any way and is then left alone, it will be set in oscillation and the oscillations will get impressed on the surrounding electromagnetic field, so that their frequencies may be observed with a spectroscope. Now whatever the laws of force governing the equilibrium, one would expect to be able to include the various frequencies in a scheme comprising certain fundamental frequencies and their harmonics. This is not observed to be the case. Instead, there is observed a new and unexpected connection between the frequencies, called Ritz's Combination Law of Spectroscopy, according to which all the frequencies can be expressed as differences between certain terms, the number of terms being much less than the number of frequencies. This law is quite unintelligible from the classical standpoint.
By terms, the spectroscopy community eventually meant 'energy levels' characterized not solely by Bohr's principal quantum number, but by the quantum numbers associated with the angular momentum of the state, yielding what are now called 'spectroscopic terms' (e.g. $^1 S_0$ for the ground state of Helium, a 'singlet' state). The energy levels, as we found for spin states of the electron, form a discrete set of states. How the frequencies are decrypted from the 'difference between ... terms' is concisely expressed using Planck's quantum hypothesis below in equation 1. The energy difference between two stationary (eigen)states is carried off by a single light quanta which we've been calling a photon, in a spectacular confirmation of the conservation of energy, \begin{equation}
E_{ph} = \frac{hc}{\lambda_{ik}} \equiv E_i - E_k.\end{equation}
The subscripts refer to different energy terms or levels, where for emission, $E_i > E_k$, and where now the wavelength (and frequency) of the emitted photon is no longer related to the energy of any single state (as is the case classically), but two states.
This is profoundly unclassical, and truly takes some getting used to. If I may try your patience with one more paragraph, I think the impact of Bohr's model must be brought out a little more sharply. Bohr's explanation of spectral lines was more to the purpose of explaining why atoms are stable at all, something that became a great scandal in physics once the meaning of Rutherford's discovery of the nucleus had come home to physics community. His, Bohr's, adoption of Planck's quantum hypothesis seems now inevitable: how else can energy be conserved unless the light-quanta's energy be associated with a change in energy of the atom? How else can one get at the allowed eigenstates unless the light quantum accounts for the energy difference between those allowed states, and the difference between their angular momenta? However logical in hindsight Bohr's great step appears to us, we cannot ignore what 'classical physicists' felt they were being asked to give up in order to access the explanatory power of the new quantum theory. They were unsubtly being asked to give up determinism, causality. I need to quote an historian to buttress my assertion here:
Of particular interest is a letter of 20 March 1913 in which
Rutherford commented on Bohr's manuscript for the Philosophical Magazine.
Apart from some minor criticism, Rutherford complained that it was 'very
difficult to form a physical idea' of the basis of Bohr's theory, a complaint that
would soon be repeated by other British physicists. More specifically, Rutherford
referred to what he called 'one grave difficulty,' namely this: 'How does an
electron decide what frequency it is going to vibrate at when it passes from one
stationary state to the other? It seems to me that you would have to assume that
the electron knows beforehand where it is going to stop.' Rutherford instinctly [sic]
sensed the element of acausality associated with Bohr's atom, a feature which
would only move to the forefront of discussion several years later. [quoted from Helge Kragh, 'The early reception of Bohr's Atomic Theory (1913-1915)', Research
Publications on Science Studies 9. Aarhus: Centre for Science
Studies, University of Aarhus, (2010)]
Note Rutherford's implicit association of the spectral line to the 'physical' vibrations of an electron whose flight had abruptly changed. This is how one of the greatest physicists kicked and screamed at quantum jumps (others later would be more shrill). Here he wants the photon's light to be associated with deterministic, causal, classical motion. Bohr's great leap is this quantum jump whereby the electron's energy change appears as a photon, all in one go, in a jump. Rutherford is having trouble with what 'physical' even means. He, like others would after him, was having a kind of vertigo, teetering between not knowing what the word meant anymore and not being willing to altar what the word had meant before Bohr's paper. Can you see why Bohr's papers were so important? Important papers make us have to look at the world in a new way. Bohr's contribution here certainly did this. And Rutherford was quite right. It meant accepting the fundamental acausality of the process that created the photon. The best one could ever do was to calculate its probability. A war was then brewing about what the new theory meant. Against its import, against the underpinnings of the new physics, against acausality, its combatants would include Einstein, Schrodinger, and de Broglie (as ironic as that sounds), and as we have seen, Rutherford (to some extent). Later, there would be Bohm, then Bell, and now there are too many to count. They stand against Heisenberg, Born, and of course Bohr (this notion of a 'Copenhagen' Interpretation hides the other 'cities' that helped to frame the banner so named, including Göttingen, Cambridge, Moscow, and Pasadena, among others). Let me end this 'paragraph' with a final assertion of my own. I do not think that the scientific revolution visited upon humanity by the crisis that is quantum mechanics is over and done. This, our current situation in the 21st century, is what revolutionary times look like. We are living in a scientific revolution right now.
References:
W. Grotrian, *Graphische Darstellung der Spektren von Atomen Ionen mit ein, zwei und drei Valenzelectronen*, Vol II. Julius Springer, Berlin, 1928. With help from Google and other sources, the title is ``Graphical representation of the spectra of atoms and ions with one, two and three valence electrons''
P.A.M. Dirac, *The Principles of Quantum Mechanics*, 4th Ed. Oxford Science Publication, Oxford University Press.
