Quantum Chemistry/Example 1

Find ⟨x⟩, ⟨x2⟩, ⟨px⟩ and ⟨px2⟩ for a quantum harmonic oscillator in the ground state, then determine the uncertainty on the position and momentum. Is the product of the uncertainty on position and momentum consistent with the Heisenberg's Uncertainty Principle? :::::::::::::::::::::::: Heisenberg's Uncertainty Principle 6 3 black white The wavefunction of a quantum harmonic oscillator in the ground state is: Using this wavefunction the average position and the average of the square of the position can be calculated. The average position: :::::: use 1 black The average square of the position: :::::::::: use 1 black :::::::::: use and 1 black The uncertainty on the position: The average momentum: ::::::::: use 1 black The average square of the momentum: ::::::::::::: use 1 black :::::::::: use 1 black :::::::::: use and 1 black The uncertainty on the momentum: The product of the uncertainty on the position and the uncertainty on the momentum is: This is equal to , therefore, a quantum harmonic oscillator in the ground state is consistent with the Heisenberg Uncertainty Principle.