University:
Carleton University
Course:
CHEM 1001 | General Chemistry I
Academic year:

2024
Views:

101

Pages:

2
Author:

ali

Review of Quantum Mechanics Classical Harmonic Oscillator Revisited Suppose the only thing we know about a classical oscillator is its energy E. Then position and momentum are uncertain with uncertainties Δx and Δp. One can show ΔxΔp = E/2π where E can have any positive value. Phase Space Plot phase space plot with a circle representing the uncertainty region Quantum Harmonic Oscillator Also have uncertainties given only energy E and ΔxΔp = E/2π but must be one of En = (n + 1/2)ħω so ΔxΔp = (n + 1/2)ħ. The position and momentum of a quantum mechanical particle cannot be determined simultaneously. Heisenberg Uncertainty Principle: The Heisenberg Uncertainty Principle states that the product of the uncertainties in position and momentum must be greater than ħ/2, where ħ = h/2π. The constant h is called Planck's constant. ΔxΔp ≥ ħ/2 - As a consequence of the uncertainty principle, position and momentum cannot be used to specify the state of a quantum mechanical particle. - The state of a quantum mechanical particle is specified by its wavefunction Ψ(x,t) at any time t. - The probability of finding the particle with position x = (x,y,z) in the interval from x to x + dx, y to y + dy, z to z + dz is then given by |Ψ(x,t)|² dxdydz at time t.