Physics
Stern Gerlach Experiment
The Stern Gerlach experiment is a fundamental physics experiment that demonstrates the quantization of angular momentum. It involves passing a beam of particles through an inhomogeneous magnetic field, which causes the particles to be deflected in a way that reveals their intrinsic angular momentum, or spin, as being quantized in specific directions. This experiment provided crucial evidence for the existence of quantized spin states in particles.
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7 Key excerpts on "Stern Gerlach Experiment"
eBook - PDFQuantum Mechanics
A Paradigms Approach
- David H. McIntyre(Author)
- 2022(Publication Date)
- Cambridge University Press(Publisher)
By using a simple example, we can focus on the principles and the new mathematics, rather than having the complexity of the physics obscure these new aspects. 1.1 STERN-GERLACH EXPERIMENT In 1922 Otto Stern and Walther Gerlach performed a seminal experiment in the history of quantum mechanics. In its simplest form, the experiment consisted of an oven that produced a beam of neu- tral atoms, a region of space with an inhomogeneous magnetic field, and a detector for the atoms, as depicted in Fig. 1.1. Stern and Gerlach used a beam of silver atoms and found that the beam was split into two in its passage through the magnetic field. One beam was deflected upwards and one down- wards in relation to the direction of the magnetic field gradient. To understand why this result is so at odds with our classical expectations, we must first analyze the experiment classically. The results of the experiment suggest an interaction between a neutral parti- cle and a magnetic field. We expect such an interaction if the particle possesses a magnetic moment M. The potential energy of this interaction is E = - M~ B, which results in a force F = 1M~ B2 . In the 2 Stern-Gerlach Experiments Stern-Gerlach experiment, the magnetic field gradient is primarily in the z-direction, and the resulting z-component of the force is F z = 0 0z 1M~ B2 (1.1) m z 0B z 0z . This force is perpendicular to the direction of motion and deflects the beam in proportion to the com- ponent of the magnetic moment in the direction of the magnetic field gradient. Now consider how to understand the origin of the atom’s magnetic moment from a classical view- point. The atom consists of charged particles, which, if in motion, can produce loops of current that give rise to magnetic moments. A loop of area A and current I produces a magnetic moment m = IA (1.2) in MKS units.
eBook - ePub- A.P. French(Author)
- 2018(Publication Date)
- Routledge(Publisher)
z component of angular momentum, is indeed quantized.We have kept this discussion simple by naively modeling the magnetic atom in classical terms (except for the possibility of a discrete set of orientations). As we discuss the quantum-mechanical description of angular momentum in more detail, we shall retreat from the view of angular momentum vectors pointing in specific directions.10-3 STERN-GERLACH EXPERIMENT: DESCRIPTIVEWe pointed out earlier that the Stern-Gerlach experiment was performed with a beam of neutral silver atoms. Since it is very difficult to produce, steer, and detect a beam of neutral atoms, one might wonder why the quantization of angular momentum should not be explored with free electrons or charged atomic ions, which are easily produced and detected and may be focused and controlled by electric and magnetic fields. The main reason is that electric and magnetic forces on charged particles are typically so much larger than forces on atomic magnetic dipoles due to magnetic field gradients that the deflections we seek to study would be swamped by deflections due to electric charge (see the exercises). However, in the case of electrons there is also a general argument that proves the impossibility of splitting a beam into separate magnetic components.9Stern-Gerlach apparatusFigure 10-5a shows a schematic diagram of an atomic beam apparatus, and Figure 10-5b is a photograph of a modern apparatus used to demonstrate the Stern-Gerlach experiment.10 A basic requirement for atomic beam experiments is a good vacuum, since at atmospheric pressure atoms have free paths of only about 10~4 cm between collisions. By maintaining a low pressure of about 10−8
- Robert Kolenkow(Author)
- 2022(Publication Date)
- Cambridge University Press(Publisher)
190 8 Quantum Angular Momentum Gerlach investigated these properties using the tool of atomic/molecular beams they initiated and that has been responsible ever since for important discoveries including several Nobel Prizes. Disclosure: The author carried out his doctoral research at Harvard in the molec- ular beams laboratory of Norman Ramsey (1915–2011, Nobel laureate in physics 1989). 8.2.1 The Apparatus As shown in the schematic dia- gram, the apparatus of the Stern– Gerlach experiment consisted of an electrically heated oven that vaporized the material to be studied. The atoms passed out of the oven through a small orifice and then through a series of slits to define a narrow beam. After passing between the poles of an electromagnet, the atoms deposited on a glass plate. The apparatus was enclosed in an evacuated chamber at a pressure so low (10 5 mm Hg) that atoms in the beam had little probability of colliding with residual gas molecules in their passage. The 1922 paper shows in their Fig. 2 a deposit of silver atoms with the electromagnet turned off to demonstrate the slit geometry. The deposit in the magnified photo was only 1.1 mm high and less than 0.1 mm wide. A run time of 5 hours was required because of the beam’s low intensity. Story has it that sulfur in the smoke from Stern’s cigar converted the metallic silver to more visible black silver sulfide, but their 1924 paper said a chemical solution was used to develop the trace. 8.2.2 Magnetic Moment in a Magnetic Field A finite region of circulating currents gives rise to a magnetic moment vector that characterizes the magnetic field generated by the currents. As an example, a flat cir- cular loop of radius r carrying current I produces a magnetic moment D I r 2 with normal to the plane of the loop. If the current is due to an electron e circulating with speed v, then I D e v 2r . The electron has angular momentum L D mvr along the line of .
eBook - PDF- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
How many projections of the electron’s orbital magnetic dipole moment on a z axis are allowed? Figure 40-7 A classical model showing the total angular momentum vector J → and the effective magnetic moment vector μ → eff . z eff,z eff J z J μ μ Figure 40-6 The allowed values of S z and μ z for an electron. s,z = – B z s = 1 _ 2 S = h 3 __ 2 S z = + h 1 _ 2 s,z = + B s S z = – h 1 _ 2 S μ μ μ μ μ 1078 CHAPTER 40 ALL ABOUT ATOMS 40-2 THE STERN–GERLACH EXPERIMENT 40.20 Sketch the Stern–Gerlach experiment and explain the type of atom required, the anticipated result, the actual result, and the importance of the experiment. 40.21 Apply the relationship between the magnetic field gradient and the force on an atom in a Stern–Gerlach experiment. ● The Stern–Gerlach experiment demonstrated that the mag- netic moment of silver atoms is quantized, experimental proof that magnetic moments at the atomic level are quantized. ● An atom with a magnetic dipole moment experiences a force in a nonuniform magnetic field. If the field changes at the rate of dB/dz along a z axis, then the force is along the z axis and its magnitude is related to the component μ z of the dipole moment: F z = μ z dB dz . Learning Objectives After reading this module, you should be able to . . . Key Ideas The Stern–Gerlach Experiment In 1922, Otto Stern and Walther Gerlach at the University of Hamburg in Germany showed experimentally that the magnetic moment of silver atoms is quantized. In the Stern – Gerlach experiment, as it is now known, silver is vaporized in an oven, and some of the atoms in that vapor escape through a narrow slit in the oven wall and pass into an evacuated tube. Some of those escaping atoms then pass through a second narrow slit, to form a narrow beam of atoms (Fig. 40‑8).
eBook - PDF- Mohsen Razavy(Author)
- 2011(Publication Date)
- World Scientific(Publisher)
We note here that each ¯ m level is split into two levels, depending on the direction of the electron spin. In our discussion we have used the first order perturbation theory to calculate the shift in the energy levels caused by an external magnetic field. This splitting, shown in Fig. 16.8, is called anomalous Zeeman effect. For an exact formulation of the problem we can write the Schr¨ odinger equation for the motion of a spinning electron in a magnetic field. In this situation the wave function has two components ψ ↑ and ψ ↓ depending on whether the spin is up or is down. In this case the Schr¨ odinger equation becomes a matrix equation i ¯ h ∂ ∂t ψ ↑ ψ ↓ = 1 2 m -i ¯ h ∇ -e c A ( r , t ) 2 · 1 ψ ↑ ψ ↓ - ge ¯ h 4 mc σ · B ( r , t ) ψ ↑ ψ ↓ + V ( r , t ) · 1 ψ ↑ ψ ↓ . (16.232) Stern–Gerlach Experiment 503 In this equation V ( r , t ) is the potential energy and 1 is the unit 2 × 2 matrix. The coupling between ψ ↑ and ψ ↓ is provided by the term σ · B = B z B x -iB y B x + iB y B z , (16.233) in (16.232). Equation (16.232) is known as the Pauli equation [34]. 16.9 Stern–Gerlach Experiment The Stern–Gerlach experiment may be regarded as one of the most important experiments in the development of quantum mechanics [35]. In this experiment a beam of hydrogen atoms is sent through an inhomogeneous magnetic field, as is shown in Fig. 16.9. The atoms are in the ground state, i.e. the electrons are in singlet S state with zero orbital angular momentum. However because of the spin magnetic moment, the beam is split into two parts of equal intensity. This splitting shows that all electrons have a magnetic moment with the same absolute value, but with two possible orientations, parallel and antiparallel to the magnetic field.
- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2023(Publication Date)
- Wiley(Publisher)
z J z J μ eff, z μ eff CHECKPOINT 40.1.1 An electron is in a quantum state for which the magnitude of the electron’s orbital angular momentum L → is 2 √ _ 3ℏ . How many projections of the electron’s orbital mag- netic dipole moment on a z axis are allowed? 40.2 THE STERN–GERLACH EXPERIMENT KEY IDEAS 1. The Stern–Gerlach experiment demonstrated that the magnetic moment of silver atoms is quantized, experimental proof that magnetic moments at the atomic level are quantized. 2. An atom with a magnetic dipole moment experiences a force in a non- uniform magnetic field. If the field changes at the rate of dB/dz along a z axis, then the force is along the z axis and its magnitude is related to the component μ z of the dipole moment: F z = μ z dB ___ dz . The Stern–Gerlach Experiment In 1922, Otto Stern and Walther Gerlach at the University of Hamburg in Germany showed experimentally that the magnetic moment of silver atoms is quantized. In the Stern–Gerlach experiment, as it is now known, silver is vapor- ized in an oven, and some of the atoms in that vapor escape through a narrow slit in the oven wall and pass into an evacuated tube. Some of those escaping atoms then pass through a second narrow slit, to form a narrow beam of atoms (Fig. 40.2.1). (The atoms are said to be collimated —made into a beam—and the second slit is called a collimator.) The beam passes between the poles of an elec- tromagnet and then lands on a glass detector plate where it forms a silver deposit. When the electromagnet is off, the silver deposit is a narrow spot. How- ever, when the electromagnet is turned on, the silver deposit should be spread vertically. The reason is that silver atoms are magnetic dipoles, and so vertical magnetic forces act on them as they pass through the vertical magnetic field of the electromagnet; these forces deflect them slightly up or down.
eBook - PDFJohn David Jackson
A Course in Quantum Mechanics
- John David Jackson, Robert N. Cahn(Authors)
- 2023(Publication Date)
- Wiley(Publisher)
Classical mechanics provides a basis for postulating quantum mechanical relations. Whether the resulting theory is correct is a matter to be tested experimentally. The results over nearly one hundred years have been entirely positive. Symmetries of the Lagrangian in classical mechanics lead to conservation laws in accordance with Noether’s theorem. The same is true in quantum mechanics, where symmetries play an even greater role, as they are represented in linear vector spaces. Dirac developed a particularly powerful representation of eigenstates of position | ′ ⟩ and momentum | ′ ⟩ , which facilitates many analyses. The Schrödinger and Heisenberg pictures provide complementary views of quantum mechan- ics. In the Schrödinger picture, the states evolve, while in the Heisenberg picture the states are fixed but the operators evolve in time. Often a physical circumstance involves not a sin- gle state, but a superposition of states; then analysis can be facilitated with a density matrix describing the mixture of states, either initial or final. John David Jackson: A Course in Quantum Mechanics, First Edition. Robert N. Cahn. © 2024 John Wiley & Sons, Inc. Published 2024 by John Wiley & Sons, Inc. Companion website: www.wiley.com/go/Jackson/QuantumMechanics 22 2 Reformulation 2.1 Stern-Gerlach Experiment The paradigm for measurement in quantum mechanics was established in 1922 when the “old quantum theory” still prevailed. In an experiment proposed by Otto Stern and executed by Walther Gerlach, a beam of neutral silver atoms was passed through a region of intense and inhomogeneous magnetic field, with the field gradient transverse to the line of flight. The force in such a region, classically, is given by F = ∇(B ⋅ ), where is the magnetic moment of the atom. The electronic configuration of the silver atom has a single s-wave electron outside a closed shell, so its magnetic moment is precisely that of the electron.
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