Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Ground State Energy of quantum oscillator

  1. Apr 5, 2014 #1
    My textbook says the ground state energy of the QSHO is given by 1/2*h_bar*w and that this is the minimum energy consistent with the uncertainty principle. However I am having trouble deriving this myself.... ΔEΔt ≥ h_bar / 2.. so then ΔE/Δfrequency ≥ h_bar / 2...

    ΔE*2*pi / w ≥ h_bar / 2
    ΔE ≥ h_bar*w / 4*pi

    what am I doing wrong?
     
  2. jcsd
  3. Apr 5, 2014 #2

    WannabeNewton

    User Avatar
    Science Advisor

    You aren't doing anything wrong per say, at least not as far as I can tell. ##\Delta t## is the characteristic time scale of the system and certainly since we have a characteristic frequency scale ##\omega##, the characteristic time scale should be given by ##\Delta t = \frac{2\pi}{\omega}##.

    I'm not entirely sure why you chose to start with the energy-time uncertainty relation; if you take a look at its derivation, particularly the assumptions about the evolution of expectation values of operators through a characteristic time period of the system, you would see that it's much more subtle a relation than the usual position-momentum uncertainty relation, but even that withstanding, the text is referring to ##\langle E \rangle ##, not ##\Delta E##, when it talks about the minimum energy consistent with the uncertainty principle. In other words ##\langle E \rangle \geq \frac{1}{2}\hbar \omega## is the desired result.

    So start instead with ##\Delta x \Delta p \geq \frac{\hbar}{2}##. Write down the expectation value for the total energy using the Hamiltonian for the QSHO and use the definition of variance and the position-momentum uncertainty relation to get an inequality for the energy expectation value involving only ##\Delta p##. Then minimize the result.
     
    Last edited: Apr 5, 2014
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Ground State Energy of quantum oscillator
Loading...