Quantum Mechanics - Theory Flashcards
Einstein proposed that an energy field radiating at frequency ν can be modeled as a collection of photons, each of energy ___________ (give the equation).
ε = hv
Where h = Planks constant = 6.626e-34 J-s
Boyd, NEGD, Pg. 54
Compton proposed that an energy field radiating at wavelength λ = c/v can be modeled as a collection of photons each having momentum ___________ (give the equation).
p = h/λ
Boyd, NEGD, Pg. 54
De Broglie proposed that a particle with energy ε and momentum p can be modeled as a wave with a frequency v = ________ and wavelength λ = ________.
v = ε/h
λ = h/p
Where h is Plank’s constant.
Body, NEGD, Pg 55
Describe a Fourier transformation.
The Fourier transform is a mathematical operation that transforms a function (often a time-domain signal) into its constituent frequencies. It’s a tool that decomposes a signal into the amplitudes of its sinusoidal components. In essence, it provides a way to move from the time domain to the frequency domain, allowing us to analyze the different frequency components of a given signal. The inverse does the opposite.
chatGPT
True or False
A takeaway from Heisenberg’s uncertainty principle is that to detect a particle, its properties must be changed.
True
Boyd, NEGD, Pg 56
State the correspondence principle.
Any quantum mechanical theory must satisfy the laws of classical physics in a regime where classical physics is valid.
Boyd, NEGD, Pg 57
What is the fundamental governing equation for quantum mechanics?
The Schrodinger Equation
Boyd, NEGD, Pg 57
The solution of the Schrodinger equation yields the _______.
Wave function
Boyd, NEGD, Pg 57
What is “degeneracy” in QM?
A situation where several energy states have the same energy level.
Boyd, NEGD, Pg 60
What does the phrase “eigenvalue problem” mean when said in relation to solving differential equations?
In the context of differential equations, the term “eigenvalue problem” generally refers to a class of problems where one seeks solutions that can be expressed as a product of two functions: one depending only on time and the other only on spatial variables. These solutions are of a special form, and the associated constants (eigenvalues) and functions (eigenfunctions) have special significance.
ChatGPT
True or False
The translational energy of a particle is quantized, and it can take on only certain clearly specified values.
True
Boyd, NEGD, Pg. 61
What is the “Hamiltonian” as it relates to classical mechanics?
The total energy of a given system.
H = KE + PE
Boyd, NEGD, Pg 62
A particle can take on different electronic energies when the electrons that surround its nucleus ____________.
Occupy different orbits.
Boyd, NEGD, Pg 68
What electronic configuration is the most stable?
The ground electronic state.
Boyd, NEGD, Pg 68
Heat transfer when radiation is a factor is governed by the _________.
(name of the relation and the equation)
Stephan Boltzmann Equation
P = σεT^4
Where
P = Power Density emitted, M/(s - m^2)
σ = Stephan-Boltzmann Constant
ε = The total emissivity
Molecular Physical Chemistry for Engineers, Yates, Pg. 39
Explain the total emissivity value (ε) as shown in the Stephan-Boltzmann law.
The value of ε is a measure of the efficiency of the reflection of radiated energy from the emitter body’s outer surface back into its interior.
A value of 1 is a perfect emitter, while 0 is a perfect reflector.
Molecular Physical Chemistry for Engineers, Yates, Pg. 39
True or False
One of the central postulates of wave mechanics is that the quantity ψ*ψ is proportional (for a given particle energy) to the number of particles that cross a unit area per unit time.
True.
Molecular Physical Chemistry for Engineers, Yates, Pg. 80
The time-independent wave function is associated with the _______ energy states of the quantum system.
Stationary
Molecular Physical Chemistry for Engineers, Yates, Pg. 82
True or False
The Schrodinger equation can be derived from other equations.
False. It is a fundamental equation which is assembled from postulates which are formulated from observation of nature.
Molecular Physical Chemistry for Engineers, Yates, Pg. 83
What is the operator for momentum? (give the equation)
\hat{p}_x = (hbar/i)∂/∂x = -ihbar(∂/∂x)
Molecular Physical Chemistry for Engineers, Yates, Pg. 83
True or False
The Schrodinger equation is an eigenvalue equation.
True
Molecular Physical Chemistry for Engineers, Yates, Pg. 85
Quantum mechanically allowed energies are referred to as _________.
Eigenstates.
Also sometimes called Eigenenergies.
Molecular Physical Chemistry for Engineers, Yates, Pg. 85
True or False
The wave function ψ, is the wave property of a particle that contains all of the information needed to describe the wavelength, KE, or momentum of a particle.
True.
Molecular Physical Chemistry for Engineers, Yates, Pg. 90
How did Max Born explain the physical meaning of the wave function?
He suggested that the wavefunction was closely related to the statistical probability of finding the particle within a given differential region of space.
Molecular Physical Chemistry for Engineers, Yates, Pg. 90
From Max Born’s explanation of the physical meaning of the wave function, what does the following correspond to?
ψ*ψdx
The physical meaning of ψ*ψdx is the probability of finding the particle in the spatial range between x and x + dx.
Molecular Physical Chemistry for Engineers, Yates, Pg. 90
Why is normalization important for wave function?
To employ the wave function usefully, it must be normalized so that it behaves in a proper statistical manner.
Molecular Physical Chemistry for Engineers, Yates, Pg. 91
True or False
Normalizing a wave function can render it unable to satisfy the Schrodinger equation.
False
Molecular Physical Chemistry for Engineers, Yates, Pg. 91
What are the four properties of a “well behaved” wave function?
- ψ must be continuous
- The derivatives of ψ must be continuous
- ψ must be a single-valued function
- ψ must not become infinite over its spatial range.
Molecular Physical Chemistry for Engineers, Yates, Pg. 93
The probability of finding a particle at a node is _______.
Zero.
Molecular Physical Chemistry for Engineers, Yates, Pg. 94
What causes a node to be present in a given quantum mechanical system?
When waves interfere, there will be nodes where complete destructive interference occurs.
Molecular Physical Chemistry for Engineers, Yates, Pg. 95
What is the expectation value of a given measurand?
The average value that we would get if we took many measurements of the same quantum system.
Molecular Physical Chemistry for Engineers, Yates, Pg. 96
The location of a particle is completely _______ when its momentum is completely certain.
Uncertain
Molecular Physical Chemistry for Engineers, Yates, Pg. 102
For the particle in an infinite-walled, one-dimensional box problem, what are the values of the wave function at the boundaries?
ψ = 0 at x = 0 and x = L
Molecular Physical Chemistry for Engineers, Yates, Pg. 112
The energy equation for the particle in the infinite-walled, one-dimensional box is given by
En = n^2h^2/(8mL^2)
Explain the physical meaning of this equation with regard to quantized energy states.
This equation tells us that only certain energies are allowed for that particle and that these eigenenergies are determined by the quantum numbers n = 1,2,3,… etc.
Molecular Physical Chemistry for Engineers, Yates, Pg. 113
In the infinite-walled, one-dimensional box problem, the lowest energy allowed corresponds to n = 1. What does this mean in regard to the lowest allowable translational energy?
The translational energy of this quantum system cannot be zero. It must have some finite and positive value as given by
En = h^2/(8mL^2)
Where n = 1
Molecular Physical Chemistry for Engineers, Yates, Pg. 114
Explain the “zero point” energy.
The lowest allowable energy state is dictated by the n = 1 quantum number and the derived energy equation from the wave function.
Molecular Physical Chemistry for Engineers, Yates, Pg. 114
What is the probability density defined as in terms of the wave function (give the equation)?
PD = ψ*ψ = abs(ψ)^2
Molecular Physical Chemistry for Engineers, Yates, Pg. 115
The probability of finding a particle at a node point is _______.
Zero.
Molecular Physical Chemistry for Engineers, Yates, Pg. 116
What is an anharmonic oscillator?
An oscillator in which k (the force constant) is not a constant value with a change in position.
Molecular Physical Chemistry for Engineers, Yates, Pg. 137
True or False
The energy levels of a quantum harmonic oscillator vary parabolicly.
False, they vary linearly with quantum number n.
En = (1/2 + n)*hv
Molecular Physical Chemistry for Engineers, Yates, Pg. 141
The energy spacing for a quantum harmonic oscillator is given as ΔE = ____
hv
Where h is Plank’s constant and v is the frequency.
Molecular Physical Chemistry for Engineers, Yates, Pg. 142
The zero point energy for a quantum harmonic oscillator is given as E_0 = ____.
hv/2
Where h is Plank’s constant, and v is the frequency.
Molecular Physical Chemistry for Engineers, Yates, Pg. 142
True or False
A quantum mechanical harmonic oscillator can penetrate beyond a classical turning point.
True.
Molecular Physical Chemistry for Engineers, Yates, Pg. 143
In QM, physical information that is accessible through a measurement is proportional to ________.
\psi^*\psi = \abs(\psi)^2
QM Full Slides
True or False:
When observing a stationary state in a QM system, the observable properties of a system do not change with time.
True.
QM Full Slides: 62
True or False:
For every observable property of a system, there exists a corresponding linear, Hermitian operator.
True
QM Full Slides: 65
True or False:
A Hermitian operator must have real eigenvalues.
True.
QM Full slides: 65
The eigenvalues are the ________in a QM system.
Observables.
QM Full Slides: 69
If two operators do not commute (their commutator does not equal zero), then there exists ____________ in their corresponding variables.
an uncertainty relationship.
QM Full slides: 72
Extra Notes: When two operators do not commute, the order of applyting the operators matters. When two operators do not commute, it usually signifies that the corresponding physical observables cannot be simultaneously measured with arbitrary precision, as per the Heisenberg uncertainty principle.
True or False
If two operators commute, then they share a common set of eigenfunctions.
True
QM Full Slides: 73
True or False:
Eigenfunctions of Hermitian operators are orthogonal.
True.
QM Full Slides: 78
True or False:
Eigenvalues of Hermitian operators can be complex.
False. Eigenvalues of Hermitian operators are always real.
The quantum mechanical harmonic oscillator has a wave function composed of __________ polynomials.
Hermite.
QM Full Slides: 92
The principle quantum number n is related to the _________ of an electron.
energy level or shell
The maximum number of electrons per shell is given by the formula _____.
2n^2
What is the angular momentum quantum number l associated with?
The total angular momentum of the electron as it moves about the nucleus.
QM Full Slides: 129
What is the magnetic quantum number (m)?
The m quantum number is associated with the component of the angular momentum along a specific axis in the atom.
In the wave function for a hydrogen atom, the Schrodinger equation can be separated into __________ and _________.
A radial part which depends only on the radial distance from the nucleus.
An angular part that depends on the angles theta and the azimuthal angle.
In the Schrodinger equation for a hydrogen atom, the radial part depends on what two quantum numbers?
n and l
In the Schrodinger equation for a hydrogen atom, the angular part of the wave function depends on what two angles?
theta = polar angle
phi = azimuthal angle
What are spherical harmonics, and how do they relate to the wave function of a hydrogen atom?
In a hydrogen atom, spherical harmonics describe the angular part of the electron’s wave function. They define the shapes and orientations of electron orbitals based on the quantum numbers ℓ and 𝑚, and they play a key role in determining the angular probability distribution of finding an electron around the nucleus.
