The proof for energy time uncertainty principle.

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Chapter 3, Problem 3.38P

To determine

The proof for energy time uncertainty principle.

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∆E ∆t ≥ ħTime is a parameter, not an observable. ∆t is some timescale over which the expectation value of an operator changes. For example, an electron's angular momentum in a hydrogen atom decays from 2p to 1s. These decays are relativistic, however the uncertainty principle is still valid, and we can use it to estimate uncertainties. The lifetime of hydrogen in the 2p state to decay to the Is ground state is 1.6 x 10-9 s. Estimate the uncertainty ∆E in energy of this excited state. What is the corresponding linewidth in angstroms?

∆E ∆t ≥ ħTime is a parameter, not an observable. ∆t is some timescale over which the expectation value of an operator changes. For example, an electron's angular momentum in a hydrogen atom decays from 2p to 1s. These decays are relativistic, however the uncertainty principle is still valid, and we can use it to estimate uncertainties. ∆E doesn't change in time, so when an excited state decays to the ground state (infinite lifetime, so no energy uncertainty), the energy uncertainty has to go somewhere. Usually, it’s in the frequency of a photon giving a width (through E = hν) to the transition line in an spectroscopy experiment. The linewidth of the 2p state in 9Be+ is 19.4 MHz. What is its lifetime? (Note: in the relativistic atom–photon system, the Hamiltonian is independent of time and both energy and its uncertainty are conserved.)

Consider a very simplistic model of atomic nucleus in 1D: a proton is completely localized in a 1D box of width L = 1.00 × 10¬14m. In other words, the proton wavefunction outside of the "nucleus" is zero. Note that L represents a typical nuclear radius. (A) What are the energies of the ground and the first excited states? If the proton makes a transition from the first excited state to the ground state, what is the angular frequency of the emitted photon? (B) What is the probability that the proton in its ground state (i.e., the lowest energy state) is not found in the distance L/12 around each boundary of the box? (C) Using the uncertainty principle, derive a minimum possible value on the momentum uncertainty in the second state above the ground state. (D) Compare your answer to the previous question (B) to probability distribution one would obtain for a classical particle. First argue about how the probability distribution would look for a classical object in its ground state. How…