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 variesby nuclear species asIassumes whole or half integer values between 0 and 8.The ¹H nucleus, a single proton, hasI =½,the lowestpossible non-zero valueallowedfor spin.
In the subatomic world governed by quantum mechanics, nuclei are better thought of as fuzzy "probability waves" rather than solid "objects".Because of theHeisenberg 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 aneigenvaluecorresponding to a particle's purespin state or eigenstate. (In German, the word "eigen" means "own" or "self").
The number of eigenstates(or purespin states)for a nucleus with spin = Iis given by:
Number of nuclear spin states = 2I + 1
Hence for the¹H nucleus withI=½, there are 2(½) + 1 = 2 possible spin states. Note that nuclei with higher values ofImay have more than a dozen spin states, but for now we will just consider the two spin states of¹H. These states are commonly denoted as ∣ +½ 〉and∣ -½ 〉often referred to as"spin-up or"parallel"andspin-downor"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 bedegenerate. 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-½silver 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. |
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. |
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! It is only when spins are subjected to a measurement process (such as being passed through the S-G apparatus), that their wave functions collapse to reveal the two separate/pure eigenstates.
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 nuclearZeeman 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. (Fabulous detailed review)[DOI Link]
Gerlach W, Stern O.Über die richtungsquantelung im magnetfeld.Ann Phys 1924;74:673-699.
As an expert in quantum mechanics and atomic physics, my knowledge is backed by an understanding of fundamental principles and empirical evidence related to spin states and energy levels of subatomic particles. To demonstrate my expertise, I'll delve into the concepts mentioned in the article regarding spin and spin states.
Spin, in the realm of quantum mechanics, refers to an intrinsic property of particles, such as electrons, protons, and neutrons, akin to their intrinsic angular momentum. This property, denoted by the symbol I, or referred to as "spin angular momentum," differs from classical angular momentum. Unlike classical objects like spinning tops, particle spin is quantized, taking on discrete values as whole or half integer multiples of Planck's constant divided by 2π.
The concept of spin state corresponds to the specific orientation or state of a particle's spin angular momentum along a chosen direction. In the context of particles with spin, the term "spin state" refers to the measurable eigenvalues associated with the spin, such as the component of angular momentum along a specified axis.
Quantum mechanics dictates that due to the Heisenberg Uncertainty Principle, we cannot precisely know the exact direction of a particle's spin at any given moment. However, we can measure certain properties, represented by eigenvalues, which correspond to the particle's pure spin state or eigenstate. These eigenstates are fundamental to understanding the behavior of spins in a quantum system.
The number of eigenstates or pure spin states for a nucleus with a specific spin (I) is given by the formula: Number of nuclear spin states = 2I + 1. For instance, for the simplest nucleus, the hydrogen nucleus (¹H) with I = ½, there exist 2(½) + 1 = 2 possible spin states, commonly denoted as ∣ +½ 〉 and ∣ -½ 〉, corresponding to "spin-up" and "spin-down" states, respectively.
The Stern-Gerlach experiment, conducted in 1922, provided compelling evidence for the quantization of angular momentum. This experiment involved passing spin-½ silver atoms through a magnetic field, resulting in the separation of these atoms into two distinct spots on a detection screen, confirming the existence of discrete spin states.
Furthermore, the physical separation observed in the Stern-Gerlach experiment also highlights an energy difference (ΔE) between the two spin states, known as the nuclear Zeeman effect. This energy difference plays a crucial role in understanding how particles in different spin states possess different energy levels in the presence of a magnetic field.
It's essential to note that while a particle may possess two distinct spin states, quantum mechanics suggests that spins typically exist in a superposition of these states until a measurement is made, causing the wave function to collapse and reveal the particle's specific spin state.
This knowledge is substantiated by various scientific references, including seminal works by Otto Stern and Walther Gerlach, theoretical contributions, and empirical evidence from experiments like the Stern-Gerlach experiment, which collectively confirm the fundamental principles governing spin and its associated states in quantum mechanics and atomic physics.
