Ch 38: Quantization

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 38: Quantization

Problem 62b

Chapter 38, Problem 62b

Consider a hydrogen atom in stationary state n. On average, an atom stays in the n = 2 state for 1.6 ns before undergoing a transition to the n = 1 state. On average, how many revolutions does the electron make before the transition?

Verified step by step guidance

Determine the orbital period of the electron in the n=2 state. The orbital period is the time it takes for the electron to complete one revolution. Use the formula for the orbital period:

T = \frac{2}{π}

, where

ω

is the angular velocity of the electron in the n=2 state.

Calculate the angular velocity

ω

using the Bohr model. The angular velocity is related to the orbital frequency

f

ω = 2 π f

. The orbital frequency for the n-th state is given by

f = \frac{v}{r}

, where

v

is the electron's velocity and

r

is the radius of the orbit.

Substitute the expressions for

v

and

r

from the Bohr model. The velocity is

v = \frac{e}{h} * \frac{k}{n}

, and the radius is

r = n ² a ₀

, where

a ₀

is the Bohr radius.

Once the orbital period

T

is calculated, determine the number of revolutions the electron makes before transitioning. This is given by

N = \frac{t}{T}

, where

t

is the average time the electron spends in the n=2 state (1.6 ns).

Perform the division to find the number of revolutions

N

. Ensure that the units of time are consistent (convert nanoseconds to seconds if necessary) before performing the calculation.

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Key Concepts

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

Quantum States

In quantum mechanics, a stationary state refers to a specific energy level of an atom where the electron's probability distribution does not change over time. For hydrogen, these states are denoted by quantum numbers, with 'n' representing the principal quantum number. The electron in a stationary state exhibits quantized energy levels, which are crucial for understanding transitions between states.

Orbital Period

The orbital period is the time it takes for an electron to complete one full revolution around the nucleus of an atom. For a hydrogen atom in a given stationary state, this period can be calculated using the principles of circular motion and the quantization of angular momentum. Understanding the orbital period is essential for determining how many revolutions occur during the time the electron remains in a specific state.

Transition Probability

Transition probability refers to the likelihood of an electron moving from one energy state to another, influenced by factors such as time and the nature of the states involved. In this context, the average time spent in a stationary state (1.6 ns for n=2) can be used to calculate the expected number of revolutions before a transition occurs. This concept is fundamental in quantum mechanics, particularly in understanding atomic behavior and emission spectra.

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Probability Distribution Graph

Related Practice

Textbook Question

Draw an energy-level diagram, similar to Figure 38.21, for the He+ ion. On your diagram: Show the ionization limit.

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Textbook Question

Draw an energy-level diagram, similar to Figure 38.21, for the He+ ion. On your diagram: Show the first five energy levels. Label each with the values of n and E_n.

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Textbook Question

Draw an energy-level diagram, similar to Figure 38.21, for the He+ ion. On your diagram: Show all possible emission transitions from the n = 4 energy level.

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Textbook Question

What is the difference in the wavelengths emitted in a 199→2 transition and a 200→2 transition?

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Textbook Question

The first three energy levels of the fictitious element X were shown in Figure P38.54. An electron with a speed of 1.4×10⁶ m/s collides with an atom of element X. Shortly afterward, the atom emits a photon with a wavelength of 1240 nm. What was the electron’s speed after the collision? Assume that, because the atom is much more massive than the electron, the recoil of the atom is negligible. Hint: The energy of the photon is not the energy transferred to the atom in the collision.

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Textbook Question

What wavelength photon does a hydrogen atom emit in a 200→199 transition?

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