Q6:4 We showed in the Lecture notes that the normalised energy eigenfunction for the lowest energy eigenvalue of the harmonic oscillator (E = ħw/2) is 1/2 exp a²r² to (x) = 2 where a = . As we encountered in the group work, in quantum mechanics the uncertainty in the position of the particle may be described by Ar = / (x²) – (x)², for a particle represented by a given wavefunction. Calculate Ax for the wavefunction yo (x). In addition, calculate the uncertainty in the momentum of the particle for the same wavefunction, Ap, namely Ap = / (p²) – (p)² and hence show that ArAp = (This is an example of a Heisenberg uncertainty relation) Note that the operator p= -iħ and p² = –ħ² . [You may assume that ſ y² exp (–y³)dy = Vñ/2.]

Q6:4 We showed in the Lecture notes that the normalised energy eigenfunction for the lowest energy eigenvalue of the harmonic oscillator (E = ħw/2) is 1/2 exp a²r² to (x) = 2 where a = . As we encountered in the group work, in quantum mechanics the uncertainty in the position of the particle may be described by Ar = / (x²) – (x)², for a particle represented by a given wavefunction. Calculate Ax for the wavefunction yo (x). In addition, calculate the uncertainty in the momentum of the particle for the same wavefunction, Ap, namely Ap = / (p²) – (p)² and hence show that ArAp = (This is an example of a Heisenberg uncertainty relation) Note that the operator p= -iħ and p² = –ħ² . [You may assume that ſ y² exp (–y³)dy = Vñ/2.]