Position and Momentum Measurements on the Harmonic Oscillator, and the Uncertainty Principle скачать в хорошем качестве

Position and Momentum Measurements on the Harmonic Oscillator, and the Uncertainty Principle

Quantum Chemistry Problem [Q21-06-00] ----------------------------------------- Question: (a) Evaluate ⟨x²⟩ for the harmonic oscillator and from this value obtain Δx=√[⟨x²⟩−⟨x⟩²]. (b) Evaluate ⟨pₓ²⟩ and ⟨p⟩² for the harmonic oscillator and calculate the uncertainty in the momentum Δpₓ. (c) Using ⟨x²⟩ and ⟨pₓ²⟩, find the uncertainty relation for the harmonic oscillator. ----------------------------------------- For a quantum object, unlike in classical mechanics, the precise measurement of both its position and its momentum is fundamentally impossible. The relationship between the intrinsic uncertainty in the measurement of the object's position and the uncertainty in the measurement of its momentum is known as the Uncertainty Principle. In this video, we show using the ground-state wavefunction of the harmonic oscillator, how the variance in the measurement operator x relates to the variance in the measurement operator p. This video also demonstrates how to calculate expectation values from the eigenfunctions of the Schroginer Equation. For the harmonic oscillator, the expectation values of x squared and p squared are related to each other by the expectation values of both sides of the Schrodinger Equation: ⟨pₓ²⟩/2m + m⟨x²⟩/2 = ℏω(2n+1)/2, where n, the vibrational quantum number, = 0, 1, 2, ... . For the ground state of the harmonic oscillator, n=0, one obtains the Uncertainty Principle relation: (Δx)(Δpₓ) ≥ ℏ/2. where (Δx) and (Δpₓ) are the "errors" or uncertainty in the measurements of position and momentum. However, for the excited states in the harmonic oscillator, (Δx) times (Δpₓ) will be even bigger than ℏ/2, so the uncertainty relationship should really be understood as the "minimum uncertainty relationship".