Harmonic oscillator and the HUP

In summary, to prove that a 1-d harmonic oscillator in ground state obeys the Heisenberg Uncertainty Principle, you need to compute delta P sub x and delta X using the 1-D ground-state wave function and the definition of expectation value.
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Homework Statement



Prove that a 1-d harmonic oscillator in ground state obeys the HUP by computing delta P sub x and delta X

Homework Equations



delta x = sqrt(<x^2>-<x>^2)
delta px = sqrt(<px^2>-<px>^2)

The Attempt at a Solution



I have absolutely no idea where to start with this. Maybe I'm missing something simple (doubtful - I'm a chem major getting crushed by p-chem - very little calc experience), but I have no idea how to do this and it's driving me absolutely nuts. Any help would be great.
 
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  • #2
You need:

the 1-D ground-state wave function of the harmonic oscillator.
the definition of expectation value (the angle bracketed objects are expectation values).
 

1. What is a harmonic oscillator?

A harmonic oscillator is a physical system that exhibits periodic motion around an equilibrium point. It can be described by a mathematical function known as a harmonic function, which follows a sinusoidal pattern.

2. What is the Heisenberg Uncertainty Principle (HUP)?

The Heisenberg Uncertainty Principle is a fundamental principle in quantum mechanics that states that it is impossible to simultaneously know the exact position and momentum of a particle. This means that the more precisely we know the position of a particle, the less precisely we can know its momentum, and vice versa.

3. How does the HUP relate to the harmonic oscillator?

The HUP relates to the harmonic oscillator through the concept of energy and time. As the energy of the oscillator decreases, the uncertainty in its position and momentum increases. This is because the oscillator spends more time near the turning points of its motion, where the position and momentum are less well-defined.

4. What is the significance of the HUP in quantum mechanics?

The HUP is significant in quantum mechanics because it sets a limit on the precision with which we can measure certain physical quantities. It also highlights the inherently probabilistic nature of quantum systems, where we can only predict the likelihood of a particle's position or momentum, rather than its exact value.

5. How does the HUP impact our understanding of the microscopic world?

The HUP has a significant impact on our understanding of the microscopic world, as it suggests that there are inherent limitations in our ability to measure and understand the behavior of particles at the quantum level. It also helps to explain certain phenomena, such as the stability of atoms and the uncertainty in the position of subatomic particles.

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