Calculate the energy difference between the ms=12m_{s}=\frac12 ('spin up') and ms=−12m_{s}=-\frac12 ('spin down') levels of a hydrogen atom in the 1s1s state when it is placed in a 1.451.45 T magnetic field in the negative zz-direction. Which level, ms=12m_{s}=\frac12 or ms=−12m_{s}=-\frac12, has the lower energy?

Ch 41: Quantum Mechanics II: Atomic Structure

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 41: Quantum Mechanics II: Atomic Structure

Problem 25

Chapter 41, Problem 25

Calculate the energy difference between the $m_{s} = \frac{1}{2}$ ('spin up') and $m_{s} = - \frac{1}{2}$ ('spin down') levels of a hydrogen atom in the $1 s$ state when it is placed in a $1.45$ T magnetic field in the negative $z$ -direction. Which level, $m_{s} = \frac{1}{2}$ or $m_{s} = - \frac{1}{2}$ , has the lower energy?

Verified step by step guidance

Understand the problem: The energy difference between the spin-up (ms = +1/2) and spin-down (ms = -1/2) states of an electron in a magnetic field is due to the Zeeman effect. The energy of each state is given by the formula E = -μz * B, where μz is the z-component of the magnetic moment and B is the magnetic field strength.

Recall the relationship between the magnetic moment and the spin quantum number: The magnetic moment μz is related to the spin quantum number ms by the formula μz = -g * μB * ms, where g is the g-factor (approximately 2 for an electron), μB is the Bohr magneton (9.274 × 10^-24 J/T), and ms is the spin quantum number (+1/2 or -1/2).

Substitute the expression for μz into the energy formula: The energy of a state becomes E = g * μB * ms * B. For ms = +1/2 (spin-up) and ms = -1/2 (spin-down), calculate the energy for each state using this formula.

Determine the energy difference: The energy difference ΔE between the two states is given by ΔE = E(ms = +1/2) - E(ms = -1/2). Substitute the values of ms and simplify the expression to find ΔE = g * μB * B.

Identify the lower energy state: Since the energy is proportional to ms, the ms = -1/2 (spin-down) state will have a lower energy in a magnetic field pointing in the negative z-direction. This is because the magnetic moment aligns with the field direction, minimizing the energy.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Zeeman Effect

The Zeeman Effect describes the splitting of spectral lines in the presence of a magnetic field. In the context of atomic physics, it explains how the energy levels of electrons in an atom, such as hydrogen, are affected by an external magnetic field, leading to different energy states for different spin orientations.

Magnetic Moment

The magnetic moment is a vector quantity that represents the magnetic strength and orientation of a magnetic source. For electrons, it is related to their spin and orbital motion, and it determines how the electron's energy levels shift in a magnetic field, with different orientations experiencing different energy levels.

Energy Level Calculation

The energy difference between the spin states in a magnetic field can be calculated using the formula ΔE = gμB B, where g is the Landé g-factor, μB is the Bohr magneton, and B is the magnetic field strength. This calculation allows us to determine which spin state has lower energy based on the orientation of the magnetic moment in the applied field.

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The hyperfine interaction in a hydrogen atom between the magnetic dipole moment of the proton and the spin magnetic dipole moment of the electron splits the ground level into two levels separated by $5.9 \times 10^{- 6}$ eV. Calculate the wavelength and frequency of the photon emitted when the atom makes a transition between these states, and compare your answer to the value given at the end of Section $41.5$ . In what part of the electromagnetic spectrum does this lie? Such photons are emitted by cold hydrogen clouds in interstellar space; by detecting these photons, astronomers can learn about the number and density of such clouds.

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Textbook Question

(a) If you treat an electron as a classical spherical object with a radius of $1.0 \times 10^{- 17}$ m, what angular speed is necessary to produce a spin angular momentum of magnitude $\sqrt{\frac{3}{4}} h$ ?

(b) Use $v = r ω$ and the result of part (a) to calculate the speed $v$ of a point at the electron's equator. What does your result suggest about the validity of this model?

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The energies of the $4 s$ , $4 p$ , and $4 d$ states of potassium are given in Example $41.10$ . Calculate $Z_{e f f}$ for each state. What trend do your results show? How can you explain this trend?

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The $5 s$ electron in rubidium (Rb) sees an effective charge of $2.771 e$ . Calculate the ionization energy of this electron.

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Textbook Question

A hydrogen atom in the $5 g$ state is placed in a magnetic field of $0.600$ T that is in the $z$ -direction. Into how many levels is this state split by the interaction of the atom's orbital magnetic dipole moment with the magnetic field?

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Textbook Question

A hydrogen atom in a particular orbital angular momentum state is found to have $j$ quantum numbers $\frac{7}{2}$ and $\frac{9}{2}$ . If $n equals 5$ , what is the energy difference between the $j = \frac{7}{2}$ and $j = \frac{9}{2}$ levels?

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