Quantum Mechanics wavefuction collapse

[sty2004]

Homework Statement

Consider a harmonic oscillator. It is in the ground state. Momentum is
measured and is found to be between 0.2 [tex]\sqrt{}m\omega h[/tex]and 1.5 [tex]\sqrt{}m\omega h[/tex]. (h here is h bar)
Energy is now immediately measured. What is the probability that the energy is
unchanged? You may want to do it numerically.

Homework Equations

The Attempt at a Solution

I don't know what to start with. Maybe P(E unchange)=|<[tex]\varphi [/tex]₀|wave after measurement>|²

 

[vela]

That's right. If the energy is unchanged, it's in the ground state, so you want to find the probability the oscillator is in the ground state after the momentum is measured.

 

[sty2004]

how to find wave after measurement then?

 

[vela]

Making a measurement causes the wave function to collapse. How does the collapse work in general?

 

[paranormal]

=|<\varphi 0|e^{-iEt/h}|2=1. This means that the probability of the energy being unchanged is 100%. However, this is just a rough estimation and a more accurate calculation would require solving the Schrodinger equation for the new wavefunction after the momentum measurement and then calculating the overlap with the initial ground state wavefunction. This would give a more precise probability for the energy being unchanged. Additionally, the values for the momentum measurement should be converted to energy units using the relation p=\sqrt{2mE}. Overall, the concept of wavefunction collapse in quantum mechanics is complex and requires a thorough understanding of the theory to accurately calculate probabilities.