Today in #PHY531 #QuantumMechanics: Linear combinations of spherical harmonics yield angular momenta with various probabilities, like this Physics GRE question. The radial equation of the hydrogen atom yields a time-independent Schrödinger equation with an effective potential.
November 4, 2026 at 9:32 PM
Today in #PHY531 #QuantumMechanics: Linear combinations of spherical harmonics yield angular momenta with various probabilities, like this Physics GRE question. The radial equation of the hydrogen atom yields a time-independent Schrödinger equation with an effective potential.
Today in #PHY531 #QuantumMechanics: In spherical coordinates, if the potential only depends on the distance from the origin, the time-independent Schrödinger is separable. The azimuthal solution is a complex exponential, and the polar solution is an associated Legendre function.
November 2, 2026 at 10:41 PM
Today in #PHY531 #QuantumMechanics: In spherical coordinates, if the potential only depends on the distance from the origin, the time-independent Schrödinger is separable. The azimuthal solution is a complex exponential, and the polar solution is an associated Legendre function.
Yesterday in #PHY531 #QuantumMechanics: Observables are Hermitian operators. The determinate states of an observable are its eigenfunctions, and measurement results are its eigenvectors. The product of the standard deviations of two observables are related to their commutator.
October 29, 2026 at 5:52 PM
Yesterday in #PHY531 #QuantumMechanics: Observables are Hermitian operators. The determinate states of an observable are its eigenfunctions, and measurement results are its eigenvectors. The product of the standard deviations of two observables are related to their commutator.
One of my #PHY531 #QuantumMechanics students drew Vector, the villain from Despicable Me, on his homework.
October 26, 2026 at 8:00 PM
One of my #PHY531 #QuantumMechanics students drew Vector, the villain from Despicable Me, on his homework.
Today in #PHY531 #QuantumMechanics: In an inner product, to move an operator from one function to the other, it must be replaced by its hermitian conjugate or adjoint, denoted by a dagger. For matrices, the hermitian conjugate is the conjugate transpose.
October 26, 2026 at 8:14 PM
Today in #PHY531 #QuantumMechanics: In an inner product, to move an operator from one function to the other, it must be replaced by its hermitian conjugate or adjoint, denoted by a dagger. For matrices, the hermitian conjugate is the conjugate transpose.
Today in #PHY531 #QuantumMechanics: We went from real vectors to complex vectors to wave functions, which live in Hilbert space. In Dirac notation, column vectors are kets, their conjugate transposes are row vectors called bras, and their inner products are brackets.
October 24, 2026 at 4:06 AM
Today in #PHY531 #QuantumMechanics: We went from real vectors to complex vectors to wave functions, which live in Hilbert space. In Dirac notation, column vectors are kets, their conjugate transposes are row vectors called bras, and their inner products are brackets.
Today in #PHY531 #QuantumMechanics: The second midterm, covering the harmonic oscillator (power series and raising/lowering operator methods), free particle, delta function potentials, finite square potentials, and step potentials.
October 21, 2026 at 5:55 PM
Today in #PHY531 #QuantumMechanics: The second midterm, covering the harmonic oscillator (power series and raising/lowering operator methods), free particle, delta function potentials, finite square potentials, and step potentials.
Last time in #PHY531 #QuantumMechanics: The scattering states of a delta function well have the same reflection and transmission coefficients as the delta function well. It has one bound state corresponding to decaying exponentials on each side.
October 7, 2026 at 4:15 PM
Last time in #PHY531 #QuantumMechanics: The scattering states of a delta function well have the same reflection and transmission coefficients as the delta function well. It has one bound state corresponding to decaying exponentials on each side.
Today in #PHY531 #QuantumMechanics: (1) The group velocity of a wave packet equals the classical particle's speed, while the phase velocity is half. (2) Bound states have normalizable eigenfunctions indexed by discrete n. Scattering states are unnormalized w/ continuous k.
September 30, 2026 at 8:09 PM
Today in #PHY531 #QuantumMechanics: (1) The group velocity of a wave packet equals the classical particle's speed, while the phase velocity is half. (2) Bound states have normalizable eigenfunctions indexed by discrete n. Scattering states are unnormalized w/ continuous k.
Today in #PHY531 #QuantumMechanics: The initial wave function can be expressed as weighted sum of the eigenfuctions of the Hamiltonian, which for the free particle is an integral and inverse Fourier transform. So, the weights are the Fourier transform of the initial state.
September 28, 2026 at 7:41 PM
Today in #PHY531 #QuantumMechanics: The initial wave function can be expressed as weighted sum of the eigenfuctions of the Hamiltonian, which for the free particle is an integral and inverse Fourier transform. So, the weights are the Fourier transform of the initial state.
Today in #PHY531 #QuantumMechanics:
Student: "God, please give me a sine."
Me: "Best He can do is a cosine."
Student: "God, please give me a sine."
Me: "Best He can do is a cosine."
September 28, 2026 at 6:02 PM
Today in #PHY531 #QuantumMechanics:
Student: "God, please give me a sine."
Me: "Best He can do is a cosine."
Student: "God, please give me a sine."
Me: "Best He can do is a cosine."
Today in #PHY531 #QuantumMechanics: For the quantum harmonic oscillator, the raising and lower operators are Hermitian conjugates of each other. Using this, we can determine the normalization constant when finding eigenfunctions by raising and lowering, resulting in this ladder:
September 25, 2026 at 7:19 PM
Today in #PHY531 #QuantumMechanics: For the quantum harmonic oscillator, the raising and lower operators are Hermitian conjugates of each other. Using this, we can determine the normalization constant when finding eigenfunctions by raising and lowering, resulting in this ladder:
Today in #PHY531 #QuantumMechanics: The Hamiltonian for the quantum harmonic oscillator can be written in terms of raising and lower operators. Applying them to an eigenfunction yields a new eigenfunction with eigenenergy ℏω higher or lower. Lowering the ground state yields 0.
September 23, 2026 at 8:12 PM
Today in #PHY531 #QuantumMechanics: The Hamiltonian for the quantum harmonic oscillator can be written in terms of raising and lower operators. Applying them to an eigenfunction yields a new eigenfunction with eigenenergy ℏω higher or lower. Lowering the ground state yields 0.
Today in #PHY531 #QuantumMechanics: The eigenfunctions of the quantum harmonic oscillator are Hermite polynomials times a Gaussian. Classically, a particle is confined to a spatial region, but quantumly, it has a nonzero probability of being found outside this region.
September 21, 2026 at 9:08 PM
Today in #PHY531 #QuantumMechanics: The eigenfunctions of the quantum harmonic oscillator are Hermite polynomials times a Gaussian. Classically, a particle is confined to a spatial region, but quantumly, it has a nonzero probability of being found outside this region.
Today in #PHY531 #QuantumMechanics: For the quantum harmonic oscillator, energy eigenfunctions are only normalizable at certain discrete energies. Its power series is unnormalizable if the series is infinite, so it terminates as a polynomial, where the order n gives E = (n+½)ℏω.
September 18, 2026 at 8:09 PM
Today in #PHY531 #QuantumMechanics: For the quantum harmonic oscillator, energy eigenfunctions are only normalizable at certain discrete energies. Its power series is unnormalizable if the series is infinite, so it terminates as a polynomial, where the order n gives E = (n+½)ℏω.
Today in #PHY531 #QuantumMechanics: We began solving the quantum harmonic oscillator using the familiar power series method. After, we'll use a new algebraic method. They will yield the same results here, but when we do angular momentum, they won't!
September 17, 2026 at 1:58 AM
Today in #PHY531 #QuantumMechanics: We began solving the quantum harmonic oscillator using the familiar power series method. After, we'll use a new algebraic method. They will yield the same results here, but when we do angular momentum, they won't!
Today in #PHY531 #QuantumMechanics: The first midterm, covering the statistical nature of the wave function through the particle in a box. Here's the first question:
September 14, 2026 at 7:26 PM
Today in #PHY531 #QuantumMechanics: The first midterm, covering the statistical nature of the wave function through the particle in a box. Here's the first question:
Today in #PHY531 #QuantumMechanics: A particle trapped in a box with initial wave function equal to an upside-down parabola has a probably density that evolves as shown in the animation.
September 9, 2026 at 7:57 PM
Today in #PHY531 #QuantumMechanics: A particle trapped in a box with initial wave function equal to an upside-down parabola has a probably density that evolves as shown in the animation.
Today in #PHY531 #QuantumMechanics: Say a particle in a box has an initial wave function that's an upside-down parabola. Expressing it as a linear combination of the eigenstates of the Hamiltonian, which are sines, is finding its Fourier series.
September 5, 2026 at 4:31 AM
Today in #PHY531 #QuantumMechanics: Say a particle in a box has an initial wave function that's an upside-down parabola. Expressing it as a linear combination of the eigenstates of the Hamiltonian, which are sines, is finding its Fourier series.
One of my students drew a picture on their homework. #PHY531 #QuantumMechanics
September 4, 2026 at 7:44 PM
One of my students drew a picture on their homework. #PHY531 #QuantumMechanics
Yesterday in #PHY531 #QuantumMechanics: The stationary states of a particle in a box are sine functions, which go to zero at the ends of the box. Their various wavelengths give rise to different energies, which are spaced quadratically apart.
September 3, 2026 at 4:14 PM
Yesterday in #PHY531 #QuantumMechanics: The stationary states of a particle in a box are sine functions, which go to zero at the ends of the box. Their various wavelengths give rise to different energies, which are spaced quadratically apart.
Today in #PHY531 #QuantumMechanics: Schrödinger's equation is a PDE. If the potential is time independent, it separates into two ODEs. One yields a complex exponential in time. The other is the time-independent Schrödinger equation, an eigenvalue relation for the Hamiltonian.
August 31, 2026 at 9:26 PM
Today in #PHY531 #QuantumMechanics: Schrödinger's equation is a PDE. If the potential is time independent, it separates into two ODEs. One yields a complex exponential in time. The other is the time-independent Schrödinger equation, an eigenvalue relation for the Hamiltonian.
Today in #PHY531 #QuantumMechanics: The expected value of a quantity Q(x,p) is found by replacing p with its operator, sandwiching Q between Ψ* and Ψ, and integrating. The spread of position and momentum measurements of identically prepared systems satisfy Heisenberg's uncertainty principle.
August 28, 2026 at 9:10 PM
Today in #PHY531 #QuantumMechanics: The expected value of a quantity Q(x,p) is found by replacing p with its operator, sandwiching Q between Ψ* and Ψ, and integrating. The spread of position and momentum measurements of identically prepared systems satisfy Heisenberg's uncertainty principle.
I've been using a refillable dry erase marker for #PHY531 #QuantumMechanics, which only last 1.75 lectures. I mentioned bringing a second marker, and a student said, "It's faster to switch to your secondary than refill your primary." That earned a laugh and a tweet.
August 28, 2026 at 6:51 PM
I've been using a refillable dry erase marker for #PHY531 #QuantumMechanics, which only last 1.75 lectures. I mentioned bringing a second marker, and a student said, "It's faster to switch to your secondary than refill your primary." That earned a laugh and a tweet.