@thomaswong.netLast 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.
@thomaswong.netToday 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.
@thomaswong.netToday 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.
@thomaswong.netToday in #PHY531 #QuantumMechanics:
Student: "God, please give me a sine."
Me: "Best He can do is a cosine."
@thomaswong.netToday 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:
@thomaswong.netToday 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.
@thomaswong.netToday 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.
@thomaswong.netToday 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.
@thomaswong.netToday 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!
@thomaswong.netToday 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:
@thomaswong.netToday 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.