Ch 40: Quantum Mechanics I: Wave Functions

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 40: Quantum Mechanics I: Wave Functions

Problem 14

Chapter 40, Problem 14

An electron in a one-dimensional box has ground state energy $2.00$ eV. What is the wavelength of the photon absorbed when the electron makes a transition to the second excited state?

Verified step by step guidance

Step 1: Understand the problem. The electron is in a one-dimensional box, and its energy levels are quantized. The energy levels are given by the formula:

E_{n} = \frac{n ² h^{2}}{8 m L^{2}}

, where

n

is the quantum number,

h

is Planck's constant,

m

is the mass of the electron, and

L

is the length of the box.

Step 2: Relate the given ground-state energy to the quantum number

n

. The ground-state corresponds to

n = 1

, and its energy is given as 2.00 eV. Use this information to determine the constant factor in the energy formula:

E_{1} = \frac{h^{2}}{8 m L^{2}}

. This constant will help calculate higher energy levels.

Step 3: Calculate the energy of the second excited state. The second excited state corresponds to

n = 3

. Using the formula for energy levels, substitute

n = 3

into the equation:

E_{3} = \frac{9 h^{2}}{8 m L^{2}}

. Compare this energy to the ground-state energy to find the energy difference.

Step 4: Determine the energy of the photon absorbed during the transition. The energy difference between the second excited state (

n = 3

) and the ground state (

n = 1

) corresponds to the energy of the absorbed photon:

E = E_{3} - E_{1}

. Use the values calculated in previous steps to find this energy difference.

Step 5: Relate the energy of the photon to its wavelength using the formula:

E = \frac{hc}{λ}

, where

h

is Planck's constant,

c

is the speed of light, and

λ

is the wavelength. Rearrange the formula to solve for

λ

λ = \frac{hc}{E}

. Substitute the energy of the photon and known constants to find the wavelength.

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Key Concepts

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

Quantum Mechanics and Energy Levels

In quantum mechanics, particles such as electrons occupy discrete energy levels within a potential well, like a one-dimensional box. The energy levels are quantized, meaning that an electron can only exist in specific states with defined energies. The ground state is the lowest energy level, while excited states are higher energy levels. The difference in energy between these levels determines the energy of photons absorbed or emitted during transitions.

Photon Energy and Wavelength Relationship

The energy of a photon is directly related to its wavelength through the equation E = hc/λ, where E is the energy, h is Planck's constant, c is the speed of light, and λ is the wavelength. When an electron transitions between energy levels, it absorbs or emits a photon with energy equal to the difference between the two levels. Thus, knowing the energy change allows us to calculate the corresponding wavelength of the absorbed photon.

Energy Level Calculation in a Particle in a Box

For a particle in a one-dimensional box, the energy levels are given by the formula E_n = n²h²/(8mL²), where n is the principal quantum number, h is Planck's constant, m is the mass of the particle, and L is the length of the box. The ground state corresponds to n=1, and the second excited state corresponds to n=3. By calculating the energy for these states, we can find the energy difference that corresponds to the photon absorbed during the transition.

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Textbook Question

When a hydrogen atom undergoes a transition from the $n equals 2$ to the $n equals 1$ level, a photon with $lamda equals 122$ nm is emitted.

(a) If the atom is modeled as an electron in a one-dimensional box, what is the width of the box in order for the $n equals 2$ to $n equals 1$ transition to correspond to emission of a photon of this energy?

(b) For a box with the width calculated in part (a), what is the ground state energy? How does this correspond to the ground state energy of a hydrogen atom?

(c) Do you think a one-dimensional box is a good model for a hydrogen atom? Explain. (Hint: Compare the spacing between adjacent energy levels as a function of $n$ .)

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Textbook Question

A particle in a box is a billiard ball ( $m = 0.20$ kg) and the box has a width of $1.3$ m, the size of a billiard table. (Assume that the billiard ball slides without friction rather than rolls; that is, ignore the rotational kinetic energy.) What is the difference in energy between the $n equals 2$ and $n equals 1$ levels? Are quantum mechanical effects important for the game of billiards?

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Textbook Question

A proton is in a box of width $L$ . What must the width of the box be for the ground-level energy to be $5.0$ MeV, a typical value for the energy with which the particles in a nucleus are bound? Compare your result to the size of a nucleus — that is, on the order of $10^{- 14}$ m.

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Textbook Question

Recall that $(∣ ψ ∣^{2}) d x$ is the probability of finding the particle that has normalized wave function $ψ (x)$ in the interval $x$ to $x + d x$ . Consider a particle in a box with rigid walls at $x equals 0$ and $x = L$ . Let the particle be in the ground level and use $ψ_{n}$ as given in Eq. ( $40.35$ ) $ψ_{n} (x) = \sqrt{\frac{2}{L}} s i n [(\frac{n π x}{L})]$ where $n equals 1 comma 2 comma 3 comma dot dot dot$ .

(a) For which values of $x$ , if any, in the range from $0$ to $L$ is the probability of finding the particle zero?

(b) For which values of $x$ is the probability highest?

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Textbook Question

(a) Find the excitation energy from the ground level to the third excited level for an electron confined to a box of width $0.360$ nm.

(b) The electron makes a transition from the $n equals 1$ to $n equals 4$ level by absorbing a photon. Calculate the wavelength of this photon.

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Textbook Question

An electron is in a box of width $3.0 \times 10^{- 10}$ m. What are the de Broglie wavelength and the magnitude of the momentum of the electron if it is in (a) the $n equals 1$ level; (b) the $n equals 2$ level; (c) the $n equals 3$ level? In each case how does the wavelength compare to the width of the box?

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An electron in a one­-dimensional box has ground ­state energy 2.002.00 eV. What is the wavelength of the photon absorbed when the electron makes a transition to the second excited state?

Verified video answer for a similar problem:

Key Concepts

Quantum Mechanics and Energy Levels

Photon Energy and Wavelength Relationship

Energy Level Calculation in a Particle in a Box

An electron in a one-dimensional box has ground state energy $2.00$ eV. What is the wavelength of the photon absorbed when the electron makes a transition to the second excited state?