Help Stern-Gerlach Experiment and Spherical Harmonics

In summary, the Stern-Gerlach experiment with a beam of hydrogenic atoms in the l=1 state traveling along the y-axis and passing through inhomogeneous magnetic fields in the z- and x-directions yields three beams corresponding to different eigenstates. The probability of the beam going into each state depends on the initial selection of the Y(z, 1), Y(z, 0), or Y(z, -1) beam.
  • #1
lavenderblue
19
0
Hey, this is going to be abit long winded...

In a Stern-Gerlach experiment a beam of hydrogenic atoms in the l=1 state traveling along the y-axis is first passed through an inhomogeneous magnetic field in the z-direction to yield three beams corresponding to three eigenstates Y(z,1), Y(z,0) and Y(z,-1). One of the beams is selected and then passed through an inhomogeneous magnetic field in the x-direction. Discuss the outcome of the experiment in each of the folowing cases:

(a) The Y(z,-1) beam is selected
(b) The Y(z,0) beam is selected
(c) The Y(z,1) beam is selected

In each case compute the probability of the beam going into the states of a,b, and c.

Could you please show your working, am so confused! Thanks :)
 
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  • #2
(a) The Y(z,-1) beam is selected: In this case, the beam will be split into three components by the inhomogeneous magnetic field in the x-direction. These components will correspond to the following eigenstates: Y(x,-1/2), Y(x,+1/2), and Y(x, -3/2). The probability of the beam going into each of these states is equal to 1/9. (b) The Y(z,0) beam is selected: In this case, the beam will be split into three components by the inhomogeneous magnetic field in the x-direction. These components will correspond to the following eigenstates: Y(x, -1/2), Y(x, 0), and Y(x, +1/2). The probability of the beam going into each of these states is equal to 1/3. (c) The Y(z,1) beam is selected: In this case, the beam will be split into three components by the inhomogeneous magnetic field in the x-direction. These components will correspond to the following eigenstates: Y(x, +1/2), Y(x, +3/2), and Y(x, +5/2). The probability of the beam going into each of these states is equal to 1/9.
 

FAQ: Help Stern-Gerlach Experiment and Spherical Harmonics

What is the Stern-Gerlach Experiment?

The Stern-Gerlach Experiment is a fundamental experiment in quantum mechanics that was conducted in 1922 by Otto Stern and Walther Gerlach. It involves passing a beam of particles, typically atoms or subatomic particles, through an inhomogeneous magnetic field and observing the deflection of the particles. This experiment provided evidence for the quantization of angular momentum and spin in particles.

How does the Stern-Gerlach Experiment relate to spherical harmonics?

The Stern-Gerlach Experiment relies on the concept of spherical harmonics to explain the behavior of particles in a magnetic field. Spherical harmonics are mathematical functions that describe the orientation and energy of atomic orbitals, which are the regions where electrons are likely to be found in an atom. These functions help to explain the different energy levels and orientations of particles as they pass through the magnetic field in the experiment.

What are spherical harmonics?

Spherical harmonics are a set of mathematical functions that describe the shape and orientation of atomic orbitals in an atom. They are derived from the solution of the Schrödinger equation, which describes the behavior of particles in quantum mechanics. Spherical harmonics have the form of a product of two terms: a radial term that describes the distance from the nucleus, and an angular term that describes the orientation of the orbital. They have many applications in physics, including explaining the behavior of particles in the Stern-Gerlach Experiment.

How are spherical harmonics used in the Stern-Gerlach Experiment?

In the Stern-Gerlach Experiment, the magnetic field causes the particles to be deflected in a certain direction, depending on their spin and energy levels. This deflection can be described using spherical harmonics, as the orientation of the particles is dependent on the angular term of the function. By understanding the behavior of spherical harmonics, scientists can better understand the deflection patterns observed in the experiment and make predictions about the behavior of different particles in the magnetic field.

What is the significance of the Stern-Gerlach Experiment and spherical harmonics?

The Stern-Gerlach Experiment and spherical harmonics have significant implications in the field of quantum mechanics. They provided evidence for the quantization of angular momentum and spin in particles, which is a fundamental concept in understanding the behavior of particles at the atomic level. This experiment also showed that particles can have discrete values for their spin and energy levels, rather than continuous values. The use of spherical harmonics in this experiment has also helped to advance our understanding of atomic structure and the behavior of particles in other experiments.

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