The oscillator is “tunneling” into the classically forbidden region. This is a purely QM phenomenon! Tunneling is a general feature of QM systems, especially those with very low mass like e- and H. ψ (x) ~ sin (kx + φ) E V 0 ψ (x) ~ sin (kx) Finite barrier ψ (x) ~ e−γ x x Even though the energy is less than the barrier height, the ...
5.4: The Harmonic Oscillator Energy Levels - Chemistry LibreTexts
Jun 30, 2023 · Tunneling in Harmonic Oscillators. We should be able to calculate the probability that the quantum mechanical harmonic oscillator is in the classically forbidden region for the lowest energy state of the harmonic oscillator, the state with \(v = 0\).
Today we will briefly discuss the classical harmonic oscillator, and then lead into the quantum harmonic oscillator and its eigenfunction and eigenvalue solutions. We will draw a connection to treating the vibrations of molecular bonds. The material in today’s lecture is covered in much more detail in Chapter 5 of McQuarrie.
Let’s first focus on a simple harmonic oscillator in classical mechanics. Why? How long does one full oscillation take? when the argument of sin or cos goes from 0 to 2π, we have one period of oscillation. τ =2π= 1 ω ν .
4.4: The Harmonic Oscillator Wavefunctions ... - Chemistry …
Sep 26, 2024 · Tunneling in the Quantum Harmonic Oscillator. The observation that the wavefunctions are not zero at the classical limit means that the quantum mechanical oscillator has a finite probability of having a displacement that is larger than what is classically possible.
6.5: Quantum Mechanical Tunneling - Chemistry LibreTexts
Apr 21, 2022 · How might the wavefunctions for the position of the atom, or other particle such as an electron, with this type of potential look? A reasonable starting approximation would be to consider a harmonic oscillator wavefunction in each well.
Here is a sneak preview of what the harmonic oscillator eigenfunctions look like: (pic ture of harmonic oscillator eigenfunctions 0, 4, and 12?) Our plan of attack is the following: non-dimensionalization → asymptotic analysis → series
The oscillator is “tunneling” into the classically forbidden region. This is a purely QM phenomenon! Tunneling is a general feature of QM systems, especially those with very low mass like e- and H. Even though the energy is less than the barrier height, the …
Tunneling through a truncated harmonic oscillator potential …
Feb 1, 1990 · The sum of a one‐dimensional, truncated harmonic oscillator potential and square well of the same range, defined in the positive half‐space, serves as a convenient and instructive example for which the Schrödinger equation may have both bound‐state and continuum solutions.
The WKB approximation is a “semiclassical calculation” in quantum mechanics in which the wave function is assumed an exponential function with amplitude and phase that slowly varies compared to the de Broglie wavelength, λ, and is then semiclassically expanded.