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Chapter 8: Problem 62

Identify the different total angular momentum states (j. \(m_{i}\) ) allowed a 3 d electron in a hydrogen atom.

Short Answer

Expert verified
The total angular momentum states possible for a 3d electron in a hydrogen atom are: \((5/2, -5/2), (5/2, -3/2), (5/2, -1/2), (5/2, 1/2), (5/2, 3/2), (5/2, 5/2), (3/2, -3/2), (3/2, -1/2), (3/2, 1/2), (3/2, 3/2)\)

Step by step solution

01

Finding possible j-values

The total angular momentum quantum number, \(j\), is determined by the formula \(|l-s| \leq j \leq |l+s|\). The spin quantum number, \(s\), is always equal to 1/2 for an electron. For a 3d electron \(l=2\). Calculate j-values which fulfill the equation.
02

Identifying allowed j-values

Plugging these values into \(|l-s| \leq j \leq |l+s|\), the j values would be \(|2-1/2| \leq j \leq |2+1/2|\), which simplifies to \(3/2 \leq j \leq 5/2\). The only allowed j-value for a 3d electron is \(j=5/2\) and \(j=3/2\).
03

Finding possible \(m_{i}\) values

The magnetic quantum number, \(m_{i}\), ranges from \(-l\) to \(l\). For a 3d electron, m-values would range from \(-2\) to \(2\) but since the total angular momentum is also including the electron spin, \(m_{i}\) is going to be in the range of \(-j\) to \(j\). Therefore, \(m_{i}\) values will range from -5/2 to 5/2 (in steps of 1) for \(j=5/2\) and from -3/2 to 3/2 (in steps of 1) for \(j=3/2\).
04

Counting the total angular momentum states

The total angular momentum states (j, \(m_{i}\)) possible are: \((5/2, -5/2), (5/2, -3/2), (5/2, -1/2), (5/2, 1/2), (5/2, 3/2), (5/2, 5/2), (3/2, -3/2), (3/2, -1/2), (3/2, 1/2), (3/2, 3/2)\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quantum Mechanics
Quantum mechanics is the fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. Unlike classical physics, which deals with the macroscopic world we are familiar with, quantum mechanics operates at a scale where seemingly bizarre and counterintuitive phenomena occur.

At the heart of quantum mechanics is the concept of the wave function, which predicts the probability of finding a particle in a certain state. When measured, the particle's wave function collapses to a single state, leading to discrete, or 'quantized' outcomes, which lends the theory its name.

Key principles of quantum mechanics include wave-particle duality, quantization of energy, uncertainty principle, and superposition, which all play a role in determining the behavior of particles and dictate that physical properties like energy, position, momentum, and angular momentum are quantized.
Angular Momentum in Quantum Systems
In quantum systems, angular momentum is one of the fundamental quantities that can be quantized, meaning it can only take on certain discrete values. Angular momentum in quantum mechanics isn't just related to the motion of a particle in orbit, but it also includes a concept known as 'spin', which is intrinsic to the particle and doesn't have a classical counterpart.

The total angular momentum of a particle is described by quantum numbers. For an electron in an atom, these include the principal quantum number (), the azimuthal (or orbital) quantum number (), and the spin quantum number (). Each of these quantum numbers can only take on specific values.

The azimuthal quantum number defines the shape of the electron's orbital, and the spin quantum number relates to the electron's intrinsic spin. The combination of an electron's orbital and spin angular momenta gives rise to the total angular momentum quantum number (), which can be crucial for determining the energy levels of electrons in atoms and the resultant atomic spectra.
Magnetic Quantum Number
The magnetic quantum number, often denoted by ), is one of the four quantum numbers that describe the unique quantum state of an electron in an atom. It specifies the orientation of the orbital's angular momentum in a particular direction, usually in the presence of an external magnetic field.

In the absence of a magnetic field, these various orientations are degenerate, meaning they have the same energy. However, in a magnetic field, due to the Zeeman effect, these levels can split. The magnetic quantum number can take on any integer value between ), where is the azimuthal quantum number. For the case of a 3d electron, , which means values range from to . These values correspond to different orientations of the electron's orbit, which translates to the direction of the angular momentum vector in space.

Understanding the magnetic quantum number is important when solving the Schrödinger equation— the fundamental equation of quantum mechanics for non-relativistic particles—to analyze the energy levels and state functions of electrons in an atom.

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Most popular questions from this chapter

As is done for helium in Table 8.3 , determine for a carton atom the various states allowed according to \(\angle S\) coupling. The coupling is between carbon's two \(2 p\) electrons (its filled 2 s subshell not participating). one of which always remains in the \(2 p\) state. Consider cases in which the other is as high as the \(3 d\) level. (Note: When both electrons are in the \(2 p\), the exclusion principle Iestricts the number of states. The only allowed states are those in which \(s_{r}\) and \(\ell_{T}\) are both even or both odd.)

Two particles in a box have a total energy \(5 \pi^{2} \hbar^{2} / 2 m L^{2}\) (a) Which states are occupied? (b) Make a sketch of \(P_{S}\left(x_{1}, x_{2}\right)\) versus \(x_{1}\) for points along the line \(x_{2}=x_{1}\) (c) Make a similar sketch of \(P_{A}\left(x_{1}, x_{2}\right)\). (d) Repeat parts (b) and (c) but for points on the line \(x_{2}=L-x_{1}\). (Note: \(\left.\sin [m \pi(L-x) / L]=(-1)^{m+1} \sin \left(m \pi .0^{\prime} L\right) .\right)\)

A Simple Model: The multielectron atom is unsolvable, but simple models go a long way. Section 7.8 gives energies and orbit radii forone- electron/hydrogenlike atoms. Let us see how useful these are by considering lithium. (a) Treat one of lithium's \(n=1\) electrons as a single clectron in a one- electron atom of \(Z=3\). Find the energy and orbit radius. (b) The other \(n=1\) electron. being in the same spatial state. must have the same energy and radius, but we must account forthe repulsion between these electrons. Assuming they are roughly one orbit diameter apar, what repulsive energy would they share, and if each claims half this energy. what would be the energies of these two electrons? (c) Approximately what charge does lithium's lone valence electron orbit, and what radius and energy would it have? (d) Is i reasonable to dismiss the role of then \(=\) I electrons in chemical reactions? (e) The actual energies of lithium's electrons are about \(-98 \mathrm{eV}\) (twice, of course) and \(-5.4 \mathrm{eV}\). How good is the model? (f) Why should the model's prediction for the valence electron's energy differ in the direction it does from the actual value?

Slater Determinant: A convenient and compact way of expressing multiparticle states of antisymmetric character for many fermions is the Slater determinant. lt is based on the fact that for \(N\) fermions there must be \(N\) different individual-particle states, or sets of quantum numbers. The ith state has sparial quantum numbers (which might be \(n_{i}, \ell_{i},\) and \(m_{c i}\) ) represented simply by \(n_{t}\) and spin quanturn number \(m_{s i^{i}}\). Were it occupied by the ith particle, the state would be \(\psi_{n}\left(x_{j}\right) m_{s i}\). A column corresponds to a given state and a row to a given particle. For instance, the first column corresponds to individualparticle state \(\psi_{n}(x,) m_{3},\) where \(j\) progresses (through the rows) from particle 1 to particle \(N\). The first row corresponds to particle I. which successively occupies all individual-particle states (progressing through the columns). (a) What property of determinants ensures that the multiparticle state is 0 if any two individualparticle states are identical? (b) What property of deterninants ensures that switching the labels on any two particles switches the sign of the multiparticle state?

Write the electronis configurations for phosphorus. germanium. and cesium.
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