Ch 41: Quantum Mechanics II: Atomic Structure

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 41: Quantum Mechanics II: Atomic Structure

Problem 19a

Chapter 41, Problem 19a

A hydrogen atom in a $3 p$ state is placed in a uniform external magnetic field $\vec{B}$ . Consider the interaction of the magnetic field with the atom's orbital magnetic dipole moment. What field magnitude $B$ is required to split the $3 p$ state into multiple levels with an energy difference of $2.71 \times 10^{- 5}$ eV between adjacent levels?

Verified step by step guidance

Understand the Zeeman effect: The splitting of energy levels in the presence of a magnetic field is due to the interaction of the magnetic field with the orbital magnetic dipole moment of the atom. The energy difference between adjacent levels is given by ΔE = μ_B * B, where μ_B is the Bohr magneton and B is the magnetic field strength.

Express the Bohr magneton (μ_B): The Bohr magneton is a physical constant given by μ_B = (e * ħ) / (2 * m_e), where e is the elementary charge, ħ is the reduced Planck's constant, and m_e is the mass of the electron.

Rearrange the formula to solve for B: From the equation ΔE = μ_B * B, isolate B to get B = ΔE / μ_B. This will allow us to calculate the required magnetic field strength.

Substitute the given energy difference (ΔE = 2.71 × 10⁻⁵ eV) into the equation. Convert ΔE from electron volts to joules using the conversion factor 1 eV = 1.602 × 10⁻¹⁹ J.

Substitute the value of μ_B (approximately 9.274 × 10⁻²⁴ J/T) into the equation and calculate B. This will give the magnitude of the magnetic field required to produce the specified energy splitting.

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Key Concepts

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

Orbital Magnetic Dipole Moment

The orbital magnetic dipole moment arises from the motion of an electron in an atom, specifically its orbital angular momentum. It is a vector quantity that represents the magnetic strength and orientation of the electron's orbit. In the presence of a magnetic field, this dipole moment interacts with the field, leading to energy level splitting, known as the Zeeman effect.

Zeeman Effect

The Zeeman effect describes the phenomenon where atomic energy levels split into multiple levels when exposed to a magnetic field. This splitting occurs due to the interaction between the magnetic dipole moment of the atom and the external magnetic field, resulting in distinct energy states. The energy difference between these states can be quantified and is crucial for understanding the behavior of atoms in magnetic fields.

Energy Level Splitting

Energy level splitting refers to the division of a single energy level into multiple levels due to external influences, such as a magnetic field. The magnitude of the splitting is determined by the strength of the magnetic field and the properties of the atom, such as its magnetic dipole moment. In this context, the required field magnitude can be calculated using the energy difference between adjacent levels, which is given as 2.71 * 10^-5 eV.

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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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Textbook Question

A hydrogen atom undergoes a transition from a $2 p$ state to the $1 s$ ground state. In the absence of a magnetic field, the energy of the photon emitted is $122$ nm. The atom is then placed in a strong magnetic field in the $z$ -direction. Ignore spin effects; consider only the interaction of the magnetic field with the atom's orbital magnetic moment. How many different photon wavelengths are observed for the $2 p right arrow 1 s$ transition? What are the $m_{l}$ values for the initial and final states for the transition that leads to each photon wavelength?

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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

In a particular state of the hydrogen atom, the angle between the angular momentum vector $\vec{L}$ and the $z$ -axis is $u equals 26.6$ °. If this is the smallest angle for this particular value of the orbital quantum number $l$ , what is $l$ ?

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Textbook Question

Calculate, in units of $U$ , the magnitude of the maximum orbital angular momentum for an electron in a hydrogen atom for states with a principal quantum number of $2$ , $20$ , and $200$ . Compare each with the value of $n h$ postulated in the Bohr model. What trend do you see?

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Textbook Question

A hydrogen atom is in a $d$ state. In the absence of an external magnetic field, the states with different $m_{l}$ values have (approximately) the same energy. Consider the interaction of the magnetic field with the atom's orbital magnetic dipole moment. Calculate the splitting (in electron volts) of the ml levels when the atom is put in a $0.800$ T magnetic field that is in the $+ z$ -direction

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