Question on Intro QM pertaining to Harmonic Oscillator

In summary, a harmonic oscillator is a physical system with periodic motion and a restoring force that is proportional to the displacement from equilibrium. It is commonly used in introductory quantum mechanics to demonstrate key concepts and is related to its frequency through its potential energy. However, it is an idealized system and cannot be applied to all physical systems. The zero-point energy, which is the minimum energy a harmonic oscillator can have, is significant in describing the uncertainty of a quantum particle.
  • #1
warhammer
151
31
Homework Statement
Consider a system in a state
|ψ >=## frac{5/√50}*|φ0 > +frac{4/√50}*|φ1 > +frac{3/√50}*|φ2 >##
where |φ0 >, |φ1 > and |φ2 > are eigenstates of a harmonic oscillator
in ground, first and second excited state respectively.
(a) Find the average energy of this systemin the state |ψ >.
(b) What is the probability that |ψ > can be found in the state |φ1 >?
Relevant Equations
Avg Energy E=P(j)*E(j) (where E(n)=ℏw(n+0.5)
P(1)=## frac{|φ1|ψ>^2/<ψ|ψ > ##
Hi. I have attached a neatly done solution to the above question. I request someone to please check my solution and help me rectify any possible mistakes that I may have made.

1654354474769.png
 
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  • #2
Last line ##4^2=16##. Other than that it looks good.
 
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Likes warhammer
  • #3
kuruman said:
Last line ##4^2=16##. Other than that it looks good.
Ah yes! Sorry for that silly error, I noticed that now.

And thank you tons for your prompt guidance 🙏🏻
 

1. What is a harmonic oscillator?

A harmonic oscillator is a system that exhibits periodic motion, where the restoring force is directly proportional to the displacement from the equilibrium position. This can be seen in systems such as a mass attached to a spring or a pendulum.

2. How is the harmonic oscillator related to quantum mechanics?

In quantum mechanics, the harmonic oscillator is a fundamental model used to describe the behavior of particles in a potential well. It is used to explain the quantization of energy levels and the wave-like nature of particles.

3. What is the Schrödinger equation for a harmonic oscillator?

The Schrödinger equation for a harmonic oscillator is a second-order differential equation that describes the time evolution of a quantum mechanical system. It is given by HΨ = EΨ, where H is the Hamiltonian operator and Ψ is the wave function of the system.

4. How does the energy of a harmonic oscillator change with respect to its quantum state?

The energy of a harmonic oscillator increases with each quantum state, or energy level. This is known as energy quantization and is a result of the wave-like nature of particles in quantum mechanics.

5. Can the harmonic oscillator model be applied to real-life systems?

Yes, the harmonic oscillator model can be applied to a variety of real-life systems, such as atoms, molecules, and even macroscopic objects like pendulum clocks. It is a useful tool for understanding and predicting the behavior of these systems in the quantum world.

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