Spin States and EnergyWhat is the difference between "spin" and "spin state"?
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In previous Q&A's we have introduced the concept of spin angular momentum (I) or simply "spin", an intrinsic magnetic property possessed by nuclei and subatomic particles. We saw that spin varies by nuclear species as I assumes whole or half integer values between 0 and 8. The ¹H nucleus, a single proton, has I = ½, the lowest possible non-zero value allowed for spin.
In the subatomic world governed by quantum mechanics, nuclei are better thought of as fuzzy "probability waves" rather than solid "objects". Because of the Heisenberg Uncertainty Principle, we cannot know the exact direction of a particle's spin at any point in time. However, we can measure and know with certainty some limited properties about the spin, such as a component of its angular momentum along a single direction. Such a physically measurable quantity is known as an eigenvalue corresponding to a particle's pure spin state or eigenstate. (In German, the word "eigen" means "own" or "self").
The number of eigenstates (or pure spin states) for a nucleus with spin = I is given by:
In the subatomic world governed by quantum mechanics, nuclei are better thought of as fuzzy "probability waves" rather than solid "objects". Because of the Heisenberg Uncertainty Principle, we cannot know the exact direction of a particle's spin at any point in time. However, we can measure and know with certainty some limited properties about the spin, such as a component of its angular momentum along a single direction. Such a physically measurable quantity is known as an eigenvalue corresponding to a particle's pure spin state or eigenstate. (In German, the word "eigen" means "own" or "self").
The number of eigenstates (or pure spin states) for a nucleus with spin = I is given by:
Number of nuclear spin states = 2I + 1
Hence for the ¹H nucleus with I = ½, there are 2(½) + 1 = 2 possible spin states. Note that nuclei with higher values of I may have more than a dozen spin states, but for now we will just consider the two spin states of ¹H. These states are commonly denoted as ∣ +½ 〉and ∣ -½ 〉often referred to as "spin-up or "parallel" and spin-down or "anti-parallel" respectively. The implications of these alternate terms will become apparent shortly.
In the absence of an external magnetic field, the two separate spin states for hydrogen are not observable and said to be degenerate. If an external magnetic field (Bo) is applied, however, a quantum-field interaction occurs allowing the two separate states to be measured/revealed.
Otto Stern and Walther Gerlach performed a famous experiment demonstrating this phenomenon in 1922. Spin-½ silver atoms were boiled off from an oven, passed between the poles of a strong magnet, and then allowed to deposit on a glass plate. Instead of a single smear of silver corresponding to all possible spin orientations, exactly two silver spots were detected. The atoms had apparently sorted themselves into "spin up" and "spin down" states as they passed through the field. This is perhaps the most tangible proof of the quantization of angular momentum that can be made to appear in the macroscopic world.
Otto Stern and Walther Gerlach performed a famous experiment demonstrating this phenomenon in 1922. Spin-½ silver atoms were boiled off from an oven, passed between the poles of a strong magnet, and then allowed to deposit on a glass plate. Instead of a single smear of silver corresponding to all possible spin orientations, exactly two silver spots were detected. The atoms had apparently sorted themselves into "spin up" and "spin down" states as they passed through the field. This is perhaps the most tangible proof of the quantization of angular momentum that can be made to appear in the macroscopic world.
Memorial plaque in Frankfurt-am-Main, Germany near entrance to the building where Stern and Gerlach performed their famous experiment.
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Diagram of Stern-Gerlach experiment. Silver atoms (spin= ½) boiled from an oven and passed through a magnet deposited in two spots (instead of one), corresponding to spin-up and spin-down states separated by the magnetic field.
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It should be emphasized that just because a nucleus has two spin states, it does not mean that the individual spins reside exclusively in one state or the other. Most modern interpretations of quantum mechanics suggest that they do not. In fact, nearly all spins exist in a weighted superposition of both states simultaneously! In experimental settings (such as being passed through the S-G apparatus) their pure eigenstates can become physically manifest. Whether this is best explained by Copenhagen, Many Worlds, or other model I will leave to the philosophers of science.
The physical separation in the deposition of silver atoms by the Stern-Gerlach apparatus also reflects an energy difference (ΔΕ) between the two states. This is known as the nuclear Zeeman effect, named after Pieter Zeeman, who in 1896 had observed the splitting of optical spectral lines by a magnetic field. This topic will be discussed in more detail in the next Q&A.
References
"Introduction to Eigenstates." Wikipedia, The Free Encyclopedia.
Cresser JD. Particle spin and the Stern-Gerlach experiment (pdf). Lecture notes in quantum mechanics, Chapter 6, from: physics.mq.edu.au/~jcresser/Phys301.html (27 April 2009)
Friedrich B. Review. A century ago the Stern-Gerlach experiment ruled unequivocally in favor of quantum mechanics. Isr J Chem 2023; 63:e202300047. (This is a scholarly historical review of the events leading up to and after the S-G experiment. It should be a “must read” for every college physics professor.) [DOI Link]
Friedrich B, Herschbach D. Stern and Gerlach: How a bad cigar helped reorient atomic physics. Physics Today 2003; 56:53-59. [DOI LINK]
Gerlach W, Stern O. Über die richtungsquantelung im magnetfeld. Ann Phys 1924;74:673-699.
"Introduction to Eigenstates." Wikipedia, The Free Encyclopedia.
Cresser JD. Particle spin and the Stern-Gerlach experiment (pdf). Lecture notes in quantum mechanics, Chapter 6, from: physics.mq.edu.au/~jcresser/Phys301.html (27 April 2009)
Friedrich B. Review. A century ago the Stern-Gerlach experiment ruled unequivocally in favor of quantum mechanics. Isr J Chem 2023; 63:e202300047. (This is a scholarly historical review of the events leading up to and after the S-G experiment. It should be a “must read” for every college physics professor.) [DOI Link]
Friedrich B, Herschbach D. Stern and Gerlach: How a bad cigar helped reorient atomic physics. Physics Today 2003; 56:53-59. [DOI LINK]
Gerlach W, Stern O. Über die richtungsquantelung im magnetfeld. Ann Phys 1924;74:673-699.
Related Questions
What is spin?
How do you predict the value of nuclear spin (I) based on the number of protons and neutrons?
Where does the energy come from to keep the precession going?
What is spin?
How do you predict the value of nuclear spin (I) based on the number of protons and neutrons?
Where does the energy come from to keep the precession going?