Quantum Mechanics: Particle in a Box

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Questions and Answers

What is the significance of the particle's states being degenerate?

They have different values of nX, nY, and nZ.

They are the same physical state.

They have the same total energy. (correct)

They exist in different dimensions.

Calculate the total energy E of a particle given the states (1, 2, 1).

$\frac{4\pi^2\hbar^2}{2mL^2}$

$\frac{6\pi^2\hbar^2}{2mL^2}$

$\frac{9\pi^2\hbar^2}{2mL^2}$ (correct)

$\frac{7\pi^2\hbar^2}{2mL^2}$

What is the probability that the particle is found in the region $0 \leq x \leq L/4$ for the state (2, 1, 1)?

0.25

0.091 (correct)

0.50

0.10

Which equation represents the relationship between the energy levels of the particle in a three-dimensional box?

$E = 4\pi^2\hbar^2 \frac{n_x^2 + n_y^2 + n_z^2}{L^2}$ Signup and view all the answers

What value does the normalization constant C need to achieve?

Integrate to 1 Signup and view all the answers

How does changing the values of nX, nY, and nZ affect the total energy E?

It alters E according to their squares. Signup and view all the answers

In the equation $E = \frac{\pi^2\hbar^2(n_x^2 + n_y^2 + n_z^2)}{2mL^2}$, what does L represent?

The length of the box Signup and view all the answers

Which parameter does not affect the energy levels of the particle in a three-dimensional box?

Speed of the particle Signup and view all the answers

Considering the sine squared integral, what mathematical operation is used?

Integration Signup and view all the answers

What mathematical form describes the wave function solutions for the radial part R(r) of the hydrogen atom?

A polynomial in r combined with an exponential function Signup and view all the answers

What happens to the degeneracy of energy levels if the lengths of the sides of the box are different?

Degeneracy breaks Signup and view all the answers

Which of the following describes the periodic nature of the azimuthal solutions Φ(φ) in the context of the hydrogen atom?

Φ(φ) retains the same value for Φ(φ) and Φ(φ + 2π) Signup and view all the answers

What is required for a solution of Schrödinger’s equation for the hydrogen atom to be accepted?

The wave function must be normalizable Signup and view all the answers

In the context of quantum numbers, what do the three quantum numbers describe for the electron in a hydrogen atom?

Energy, shape, and orientation of the orbital Signup and view all the answers

How does the function of azimuth angle φ relate to the other quantum numbers in the Schrödinger equation for the hydrogen atom?

It is independent of potential energy Signup and view all the answers

Which of the following describes the degeneracy of the state with energy E1,1,1 in an equal-sided box?

6-fold degenerate Signup and view all the answers

What is a characteristic of the angular solutions Θ(θ) in the context of the hydrogen atom?

They are polynomials including powers of sin θ and cos θ Signup and view all the answers

Which term refers to different states of the hydrogen atom that share the same quantum number n but differ in l and ml values?

Degenerate states Signup and view all the answers

What is the primary characteristic of p states in a hydrogen atom?

Having l = 1 and not spherically symmetric Signup and view all the answers

In the context of transitions between energy states, when does radiation occur?

During the transition between two states with nonzero probabilities Signup and view all the answers

What happens to the expectation value of the electron's position when it transitions between energy levels?

It oscillates sinusoidally Signup and view all the answers

What do you call the probability associated with an electron in quantum state n?

Wave function Signup and view all the answers

How are the orbital shapes characterized for s, p, d states in hydrogen?

s states are spherical, p states are dumbbell-shaped, d states are more complex Signup and view all the answers

Which of the following states corresponds to the quantum number l = 2?

d state Signup and view all the answers

What does the term 'degenerate' refer to in the context of atomic states?

States with the same energy level despite different quantum numbers Signup and view all the answers

What does the principal quantum number (n) represent in the context of electron energy levels?

The quantization of energy levels Signup and view all the answers

Which letter corresponds to the angular momentum state when l = 0?

s Signup and view all the answers

In the context of the hydrogen atom, what does the probability density |Ψ|² indicate?

The probability of finding the electron in a specific region Signup and view all the answers

Which of the following statements about the average value of r for a 1s electron is correct?

It is equal to $1.5a_0$. Signup and view all the answers

What characterizes the wave functions of hydrogen atom states with l = 0?

They are spherically symmetric. Signup and view all the answers

What does the magnetic quantum number describe?

The component of angular momentum along a specified axis Signup and view all the answers

In the context of the probability of finding the electron, the equation is expressed as P(r)dr. What variable primarily influences this expression?

The radius vector r Signup and view all the answers

What happens to the angular momentum of an electron in a quantized state?

It is both conserved and quantized. Signup and view all the answers

What determines whether a transition in an atom is allowed or forbidden?

The integral is not zero Signup and view all the answers

Which changes are permitted for transitions between states of different principal quantum numbers (n)?

Both l can change by 1 or -1 and ml can stay the same Signup and view all the answers

In the semiclassical model of an H atom in an external magnetic field, how is the electron's motion characterized?

It is described by a classical circular orbit Signup and view all the answers

What characterizes the magnetic quantum number (m) in relation to the orbital quantum number (l)?

m takes values between -l and +l Signup and view all the answers

What condition must be met for an excited state of an atom to radiate electromagnetic waves?

The integral representing the transition must be non-zero Signup and view all the answers

What is the relationship between angular momentum and the orbital magnetic moment in an atom?

The orbital magnetic moment is proportional to angular momentum Signup and view all the answers

What specific change in the quantum number is unrestricted during allowed transitions in hydrogen?

The principal quantum number (n) Signup and view all the answers

Which of the following statements is true regarding the conditions for allowed transitions in a hydrogen atom?

l can change by 1 or -1 while ml can stay the same Signup and view all the answers

Study Notes

Particle in a Three-Dimensional Box

A particle can exist in three possible states (nX, nY, nZ) = (2, 1, 1), (1, 2, 1) or (1, 1, 2)
These states are degenerate, meaning they have the same total energy (E) despite differences in nX, nY, and nZ values.
The total energy is calculated as 𝐸 = 𝐸𝑋 + 𝐸𝑌 + 𝐸𝑍 = 4𝜋 2 ℏ2 / (2𝑚𝐿2) + 𝜋 2 ℏ2 / (2𝑚𝐿2) + 𝜋 2 ℏ2 / (2𝑚𝐿2) = 3𝜋 2 ℏ2 / (𝑚𝐿2)
The normalization constant (C) is calculated using the integral ∫ 𝜓 𝑥, 𝑦, 𝑧 2 𝑑𝑉 = 𝐶 2 ∫ sin2 (𝑛𝑋 𝜋𝑥 / 𝐿) 𝑑𝑥 ∫ sin2 (𝑛𝑌 𝜋𝑦 / 𝐿) 𝑑𝑦 ∫ sin2 (𝑛𝑍 𝜋𝑧 / 𝐿) 𝑑𝑧 = 1

Energy Degeneracy

The energy levels of a particle in a three-dimensional box are complex.
The first six energy levels are:
- E1,1,1 – 6-fold degenerate
- E2,1,1 – Not degenerate
- E2,2,1 – 3-fold degenerate
- E3,1,1 – 3-fold degenerate
- E2,1,2 – 3-fold degenerate
- E1,2,2 – 3-fold degenerate
When the sides of the box are different lengths, the degeneracy is broken.
The energy equation becomes 𝐸= 𝑛𝑋 2 / 𝐿𝑋 2 + 𝑛𝑌 2 / 𝐿𝑌 2 + 𝑛𝑍 2 / 𝐿𝑍 2 𝜋 2 ℏ2 / (2𝑚)

The Hydrogen Atom

The Schrödinger equation for the hydrogen atom is solved using spherical polar coordinates.
The solutions require three quantum numbers to describe the electron.
The solutions are separated into radial (R), zenith (Θ), and azimuthal (Φ) components.
The radial solution (R) is an exponential function multiplied by a polynomial in r.
The zenith solution (Θ) is a polynomial containing powers of sinθ and cosθ.
The azimuthal solution (Φ) is periodic and depends on eiml, where ml is an integer.

Quantum Numbers

Principal Quantum Number (n): n = 1, 2, 3...
Orbital Quantum Number (l): l = 0, 1, 2, 3… (s, p, d, f orbitals)
Magnetic Quantum Number (ml): −l ≤ m ≤ +l
Electron Spin Quantum Number (ms): ms = +1/2 or -1/2

The Bohr Model

The Hydrogen atom energy level formula obtained using the quantum mechanical model matches the one obtained by Bohr.

Electron Angular Momentum

The angular momentum (L) of an electron is both conserved and quantized.
The orbital kinetic energy of the electron and the magnitude of its angular momentum are related.

Electron Probability Density

The probability density for an electron in a hydrogen atom is given by |Ψ|2 = RΘΦ.
The integral of |Ψ|2 over all space is equal to 1.
The most probable value of r for a 1s electron is a0 (Bohr radius).
The average value of r for a 1s electron is 1.5a0.

Radiative Transitions

Bohr postulated that the frequency of emitted radiation is determined by the energy difference between the initial and final states.
Frequency = (Em - En) / h
Allowed Transitions: When the integral ∫ Ψn,l,ml ∗ μ Ψn’,l’,m’l dτ is not zero (finite).
Forbidden Transitions: The integral above is zero.

Selection Rules

Δl = ±1
Δml = 0, ±1

Zeeman Effect

When a hydrogen atom is placed in an external magnetic field, the energy levels split.
This splitting is called the Zeeman effect.
The splitting is due to the interaction between the magnetic moment of the electron and the magnetic field.
The energy shift is given by ΔE = -μl Bz = -mlμB Bz
The value of ml determines the splitting pattern.

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Description

Explore the principles of a particle in a three-dimensional box, including energy calculations and the concept of degeneracy. This quiz covers the three possible states of the particle and the impact of varying box dimensions on energy levels. Test your understanding of these fundamental quantum mechanics concepts.

Quantum Mechanics: Particle in a Box

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Podcast

Questions and Answers

What is the significance of the particle's states being degenerate?

Calculate the total energy E of a particle given the states (1, 2, 1).

What is the probability that the particle is found in the region $0 \leq x \leq L/4$ for the state (2, 1, 1)?

Which equation represents the relationship between the energy levels of the particle in a three-dimensional box?

What value does the normalization constant C need to achieve?

How does changing the values of nX, nY, and nZ affect the total energy E?

In the equation $E = \frac{\pi^2\hbar^2(n_x^2 + n_y^2 + n_z^2)}{2mL^2}$, what does L represent?

Which parameter does not affect the energy levels of the particle in a three-dimensional box?

Considering the sine squared integral, what mathematical operation is used?

What mathematical form describes the wave function solutions for the radial part R(r) of the hydrogen atom?

What happens to the degeneracy of energy levels if the lengths of the sides of the box are different?

Which of the following describes the periodic nature of the azimuthal solutions Φ(φ) in the context of the hydrogen atom?

What is required for a solution of Schrödinger’s equation for the hydrogen atom to be accepted?

In the context of quantum numbers, what do the three quantum numbers describe for the electron in a hydrogen atom?

How does the function of azimuth angle φ relate to the other quantum numbers in the Schrödinger equation for the hydrogen atom?

Which of the following describes the degeneracy of the state with energy E1,1,1 in an equal-sided box?

What is a characteristic of the angular solutions Θ(θ) in the context of the hydrogen atom?

Which term refers to different states of the hydrogen atom that share the same quantum number n but differ in l and ml values?

What is the primary characteristic of p states in a hydrogen atom?

In the context of transitions between energy states, when does radiation occur?

What happens to the expectation value of the electron's position when it transitions between energy levels?

What do you call the probability associated with an electron in quantum state n?

How are the orbital shapes characterized for s, p, d states in hydrogen?

Which of the following states corresponds to the quantum number l = 2?

What does the term 'degenerate' refer to in the context of atomic states?

What does the principal quantum number (n) represent in the context of electron energy levels?

Which letter corresponds to the angular momentum state when l = 0?

In the context of the hydrogen atom, what does the probability density |Ψ|² indicate?

Which of the following statements about the average value of r for a 1s electron is correct?

What characterizes the wave functions of hydrogen atom states with l = 0?

What does the magnetic quantum number describe?

In the context of the probability of finding the electron, the equation is expressed as P(r)dr. What variable primarily influences this expression?

What happens to the angular momentum of an electron in a quantized state?

What determines whether a transition in an atom is allowed or forbidden?

Which changes are permitted for transitions between states of different principal quantum numbers (n)?

In the semiclassical model of an H atom in an external magnetic field, how is the electron's motion characterized?

What characterizes the magnetic quantum number (m) in relation to the orbital quantum number (l)?

What condition must be met for an excited state of an atom to radiate electromagnetic waves?

What is the relationship between angular momentum and the orbital magnetic moment in an atom?

What specific change in the quantum number is unrestricted during allowed transitions in hydrogen?

Which of the following statements is true regarding the conditions for allowed transitions in a hydrogen atom?

Study Notes

Particle in a Three-Dimensional Box

Energy Degeneracy

The Hydrogen Atom

Quantum Numbers

The Bohr Model

Electron Angular Momentum

Electron Probability Density

Radiative Transitions

Selection Rules

Zeeman Effect

Studying That Suits You

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