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Questions and Answers
As the frequency of light increases, wavelength decreases.
As the frequency of light increases, wavelength decreases.
True (A)
All photons have the same energy.
All photons have the same energy.
False (B)
The particle in a box has ground state at n=0.
The particle in a box has ground state at n=0.
False (B)
The maximum probability density for every PIB stationary state is at the center of the box.
The maximum probability density for every PIB stationary state is at the center of the box.
Probability density can never be negative.
Probability density can never be negative.
For n=1 PIB stationary state, the probability for finding the particle in the left third of the box equals the probability for finding it in the middle third.
For n=1 PIB stationary state, the probability for finding the particle in the left third of the box equals the probability for finding it in the middle third.
The PIB energy levels are all equally spaced.
The PIB energy levels are all equally spaced.
For all QM states, Ψ(x,t) is the product of a function of x & a function of t.
For all QM states, Ψ(x,t) is the product of a function of x & a function of t.
The product of a number and its complex conjugate is always a real number.
The product of a number and its complex conjugate is always a real number.
If we measure the property of A when Ψ is not an eigenfunction of Â, then we can get a result that isn't an eigenvalue of Â.
If we measure the property of A when Ψ is not an eigenfunction of Â, then we can get a result that isn't an eigenvalue of Â.
A linear combination of two solutions to the time independent Schrodinger equation must be a solution of this equation.
A linear combination of two solutions to the time independent Schrodinger equation must be a solution of this equation.
If f is an eigenfunction of the linear operator  with eigenvalue a, then f is an eigenfunction of the linear operator Â^2 with eigenvalue a^2.
If f is an eigenfunction of the linear operator  with eigenvalue a, then f is an eigenfunction of the linear operator Â^2 with eigenvalue a^2.
All Hermitian operators are real.
All Hermitian operators are real.
If two eigenfunctions commute, they must have the same eigenfunctions.
If two eigenfunctions commute, they must have the same eigenfunctions.
If a wavefunction is simultaneously the eigenfunction of two operators, then that wavefunction will also be an eigenfunction of the product of those two operators.
If a wavefunction is simultaneously the eigenfunction of two operators, then that wavefunction will also be an eigenfunction of the product of those two operators.
Flashcards
Light Frequency and Wavelength
Light Frequency and Wavelength
Inverse relationship: as frequency increases, wavelength decreases.
Photon Energy
Photon Energy
Photons vary in energy, described by E = hν (ν = frequency).
Ground State Energy (PIB)
Ground State Energy (PIB)
The lowest energy state for a particle confined in a box is when n=1.
PIB Probability Density
PIB Probability Density
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PIB Energy Level Spacing
PIB Energy Level Spacing
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Probability Density
Probability Density
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Wavefunction Separation
Wavefunction Separation
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Complex Conjugate Product
Complex Conjugate Product
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Observable Measurement
Observable Measurement
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Linear Combination Solution
Linear Combination Solution
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Eigenfunction of ²
Eigenfunction of ²
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Hermitian Operators
Hermitian Operators
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Commuting Eigenfunctions
Commuting Eigenfunctions
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Simultaneous Eigenfunction
Simultaneous Eigenfunction
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Study Notes
Light and Photons
- Light frequency and wavelength have an inverse relationship; as frequency increases, wavelength decreases.
- Photons vary in energy, described by the equation E = hν, where ν represents the frequency; thus, not all photons possess the same energy.
Quantum Mechanics: Particle in a Box (PIB)
- The ground state energy level for a particle in a box is n=1; n=0 indicates no particle in the box.
- Maximum probability density in PIB stationary states is not at the center of the box, which actually has zero probability density.
- For the n=1 state, the probability of finding the particle in the left third of the box does not equal the probability in the middle third due to lack of symmetry in this state.
- PIB energy levels are not equally spaced; energy spacing increases with n according to the formula ΔE = (2nπh²)/(8ma²).
Wave Functions and Probability
- Probability density is always positive; it cannot be negative, even for non-real wavefunctions.
- For quantum states, the wavefunction Ψ(x, t) does not always separate into independent functions of x and t; this only occurs if the Hamiltonian is time-independent.
Eigenfunctions and Hermitian Operators
- The product of a number and its complex conjugate is always a real number.
- The measurement of an observable leads to an eigenvalue, even if the wavefunction is not an eigenfunction of the corresponding operator.
- A linear combination of two solutions to the time-independent Schrödinger equation is not guaranteed to be a solution unless both solutions share the same eigenvalues.
- If a function is an eigenfunction of the operator  with eigenvalue a, it remains an eigenfunction of the operator ² but with eigenvalue a².
- Hermitian operators are defined as equal to their complex conjugate transpose, but this does not imply that they are real.
- Commuting eigenfunctions do not necessarily share the same eigenfunctions; they can possess a mutual set of eigenfunctions.
- A wavefunction that is simultaneously an eigenfunction of two operators will also be an eigenfunction of the product of those operators.
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