(a) Using a symmetry argument rather than a calculation,...

(a) Using a symmetry argument rather than a calculation, determine the value of $p_{\mathrm{av}}$ for a simple harmonic oscillator. (b) Conservation of energy for the harmonic oscillator can be used to relate $p^{2}$ to $x^{2}$. Use this relation, along with the value of $\left(x^{2}\right)_{x}$ from Problem $24,$ to find $\left(p^{2}\right)_{a v}$ for the oscillator in its ground state. $(c)$ Using the results of $(a)$ and $(b),$ show that $\Delta p=\sqrt{\hbar \omega_{0} m / 2}$

Key Concepts

Simple Harmonic Oscillator

The simple harmonic oscillator is a foundational model in both classical and quantum mechanics. It describes systems where the restoring force is directly proportional to displacement, leading to sinusoidal oscillation. In quantum mechanics, its quantized energy levels and associated wave functions provide insight into more complex systems, and many techniques in quantum theory are illustrated using it.

Symmetry in Quantum Mechanics

Symmetry plays a critical role in quantum mechanics, particularly in determining expectation values. For example, if a system’s potential or wave function is symmetric about a point, certain observables such as momentum, which are odd functions under inversion, have an expectation value of zero. This use of symmetry can simplify calculations by bypassing explicit integrals.

Conservation of Energy in Quantum Systems

The conservation of energy principle, a cornerstone in physics, also applies to quantum mechanical systems. For systems like the harmonic oscillator, an energy relation exists between kinetic and potential contributions. This relationship allows one to connect expectation values of different observables, such as relating ?p²? to ?x²?, by setting the sum of the kinetic and potential energy equal to the quantized energy levels.

Expectation Values and Operator Methods

Expectation values in quantum mechanics provide the average results of measurements on a quantum state and are calculated using operators. Key techniques involve using known eigenstates and their properties. These averages, including those of x² and p², are essential to determine uncertainties in observables, such as the momentum spread ?p in the ground state of the oscillator.

Quantum Ground State and Uncertainty

The ground state of a quantum system, particularly in the harmonic oscillator, is the lowest energy state with a uniquely determined wave function. Calculations in this state lead to precise values for uncertainties of observables. The inherent quantum uncertainty, manifested for instance by ?p, arises from the wave-like nature of particles and is rooted in the Heisenberg uncertainty principle. This principle is characterized by the relationship between the spreads of complementary variables such as position and momentum.

Transcript

00:01 Hello, and in this question here, we want to calculate delta p for the ground state of the simple harmonic oscillator.

00:09 So this question here is a continuation of question 24, where now we're looking to calculate delta p.

00:16 Where delta p is the square root of the average value of p squared minus the average value of p all to be squared.

00:24 And in the first question, in the first part of the question, we want to calculate the average value of p.

00:29 So to do this, we'll use a symmetry argument, and we're going to let this green dot here be the point at which the particle oscillates around.

00:39 So how does the particle move? well, let's say it's going to first move this way.

00:45 Then it's going to reach some point where the potential causes a big enough force to pull the particle back towards the green part, the green dot.

00:55 And then its momentum is going to bring it past the green dot into the, towards the left until the force is until the restoring force drags the particle back towards the green dot and it's going to continue with this motion forever.

01:12 So we have a positive to the contribution to the momentum.

01:18 So we're just going to say this arrow here is a positive contribution to the momentum.

01:23 But it's going to get cancelled.

01:26 Sorry, that should be raised.

01:31 But that this positive part of the momentum here in red is going to get cancelled by a negative contribution to the momentum.

01:40 So these two red arrows will cancel each other.

01:43 And similarly, these two black arrows will cancel each other...