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What does the angular momentum quantum number ℓ describe?
What can the magnetic quantum number 𝑚ℓ value for a p-orbital (ℓ = 1) include?
What is the maximum value of ℓ for the third energy level (n = 3)?
What happens to the energy gap between orbitals as the value of n increases?
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How many degenerate orbitals are in a d-subshell (ℓ = 2)?
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Which statement about radial nodes is true when increasing the value of n?
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What is represented by the orbital surfaces showing a 90% probability of finding the electron?
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Which of the following orbital shapes corresponds to ℓ = 0?
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What does the wavefunction Ψ represent in quantum mechanics?
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According to the uncertainty principle, what happens when the position of an electron is precisely measured?
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What is represented by the square of the wavefunction, Ψ²?
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What does Schrödinger's equation allow researchers to calculate?
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In the context of quantum mechanics, what does the Hamiltonian operator (Ĥ) represent?
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What does the equation 𝜓𝑛(𝑥) = (2/L)^(0.5) sin(nπx/L) describe?
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What is a key characteristic of the wavefunctions representing electrons in quantum mechanics?
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What are nodes in a particle in a box scenario?
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What does the Hamiltonian for the PIAB system represent in the provided equation?
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What does the term zero-point energy refer to in the context of quantum mechanics?
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Which of the following statements about energy in a PIAB is true?
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What is the role of the quantum numbers n, ℓ, and mℓ in the wavefunction of an electron?
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How does moving from one dimension to three dimensions affect the wavefunction?
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What characterizes the principal quantum number n in atomic theory?
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In the wavefunction for the hydrogen atom, what do the components R(r) and Υ(θ, φ) represent?
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What is defined as 'vacuum' in the context of the energy of electrons?
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Study Notes
Determinant vs. Indeterminant
- Classical physics: By knowing an object’s initial acceleration, velocity, and position, and any additional forces felt by the object over time, you can determine the exact path that object will take.
- Classical physics: Under identical conditions, copies of that object will always follow the exact same path.
- Quantum physics: Knowing an electron’s position and velocity isn’t possible, thus we cannot know its trajectory with absolute certainty.
Wavefunctions
- Wavefunction: A mathematical description of the wavelike behavior of an electron, denoted by Ψ.
- Ψ2: Represents a probability density map (where is the electron most likely to be?)
- Ψ2 ≥ 0
- More commonly referred to as an orbital.
The Particle in a Box
- A simple example of a wavefunction describes a particle in a box:
2 𝑛𝜋𝑥
𝜓𝑛 𝑥 = sin , n = 1, 2, 3, …
𝐿 𝐿
Where L is the length of the box and x is position. - A free-moving particle that is bound on both sides by infinitely high potential walls (standing wave).
- No potential within the well itself.
- Nodes at the box walls (0 and L) and at all fractions of L/n.
Schrödinger Equation
- Schrödinger's equation: Allows us to calculate the energy of a system if we know the wavefunction that describes it. = 𝐸Ψ 𝐻Ψ
- Ĥ is the Hamiltonian operator, a set of mathematical operations that represent the combined kinetic and potential energies of a system.
- Performing those operations on Ψ will return Ψ multiplied by a real number, E.
- For the Particle in a Box system, the Hamiltonian is:
ℏ2 𝜕 2 − 2𝑚 𝜕𝑥 2
The Solution
- The solution to Schrödinger’s equation for the Particle in a Box is: 25ℎ2 ℎ2 𝑛2 8𝑚𝐿2 𝐸𝑛 = 2ℎ2 8𝑚𝐿2 𝑚𝐿2
- Key takeaways:
- Particles have a minimum energy (zero-point energy): 9ℎ2 / 8𝑚𝐿2
- Energy is inversely proportional to L: ℎ2 / 2𝑚𝐿2
- n is the first of three quantum numbers used to quantize the energy of an electron
From One Dimension to Three
- The Particle in a Box model is a simplified model that only describes an electron moving in one dimension, whereas atoms are three dimensional.
- Moving from one dimension to three greatly complicates the wavefunction.
- Must now account for coulombic potential
- Expressing the wavefunction in terms of spherical coordinates makes things a bit easier
- The solution for the Hydrogen atom looks like:
Ψ 𝑟, 𝜃, 𝜙 = 𝑅(𝑟)Υ(𝜃, 𝜙)
- 𝑅(𝑟) is known as the radial wavefunction.
- Υ(𝜃, 𝜙) is known as the angular wavefunction.
The Principal Quantum Number - n
-
First quantum number (n): Comparable to the n used in the Bohr model’s energy levels.
-
Determines the size of the orbital and the energy of the electron.
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All energies are negative. E = 0 is defined as “vacuum,” where the electron will simply leave the atom.
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As the value of n increases:
- The larger the orbital.
- The smaller the energy gap between orbitals.
-
Energy equation: 𝐸𝑛 = − 𝑚𝑒 𝑞 4 −18 𝐽 1 32𝜋 2 𝜖02 ℏ2 𝑛2 𝑛2 = -2.18 × 10
𝑛2
The Angular Momentum Quantum Number - ℓ
- Second quantum number (ℓ): Describes the shape of the orbital.
- Must be a whole number (≥ 0).
- For the nth energy level, the maximum value for ℓ is n-1 (if n = 3, ℓ = 0, 1, 2.)
- Letters for the various orbital shapes (and ℓ values):
- ℓ = 0, 1, 2, 3 → s, p, d, f (respectively)
The Magnetic Quantum Number - 𝑚ℓ
- Third quantum number (𝑚ℓ): Describes the spatial orientation of the orbital.
- Must be an integer (includes zero).
- For any orbital, the value of 𝑚ℓ can go from −ℓ to +ℓ (for a p-orbital (ℓ = 1), the values of 𝑚ℓ are -1, 0, and 1).
- Orbitals that share the same value of n and ℓ are said to be in the same subshell.
- Each subshell contains 2ℓ + 1 degenerate orbitals.
- Degenerate orbitals are energetically equivalent.
- Especially important when discussing molecular bonding because those bonds are formed by overlapping atomic orbitals.
Orbital Shapes
- 1s, 2p, and 3d orbitals.
- Orbital surfaces represent the volume in which there is a 90% probability of finding the electron.
S-Orbitals
- Probability density, Ψ 2 , corresponds directly with the orbital surface.
- Radial distribution function: Multiplying Ψ 2 by the volume of a shell at distance r from the nucleus.
- For hydrogen, the maximum at 52.9 pm corresponds with the distance predicted by Bohr.
Increasing the Value of n
- Increasing the value of n increases the “size” of the orbital.
- Introduces a feature called a radial node—zero probability of finding the electron at that distance.
P-orbitals
- p-cloud composed of three orbitals 𝑝𝑥 , 𝑝𝑦 , and 𝑝𝑧—each one goes along its corresponding Cartesian axis.
- All two-lobed p-orbitals (sometimes referred to as dumbbells) have a node at the nucleus (referred to as an angular node).
- The off-axis plane (ex. 𝑝𝑥 = yz plane) has a zero probability region.
D-orbitals
- D-orbitals have more complex shapes than s and p-orbitals.
- There are five d-orbitals per subshell—labeled 𝑑𝑥𝑦 , 𝑑𝑥𝑧 , 𝑑𝑦𝑧 , 𝑑𝑥2−𝑦2, and 𝑑𝑧2 .
- 𝑑𝑥2−𝑦2 and 𝑑𝑧2 orbitals have four lobes, all others have two lobes and a torus.
- Angular nodes become more complicated.
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Description
Explore the fundamental concepts of quantum physics, focusing on the key differences between deterministic and indeterministic behaviors in classical and quantum mechanics. This quiz also introduces wavefunctions and the particle in a box model, providing insights into electron behavior and probability mapping.
