Exercise:
abcliste abc What is the expectation value and the uncertay for the energy of an energy eigenstate of a particle in an infinite potential well? abc What is the expectation value and the uncertay for a superposition of ground state and first excited state: psixt fracsqrtpsi_x t + fracsqrtpsi_xt abcliste

Solution:
abcliste abc The expectation value for the energy of the energy eigenstate psi_nx t is langle hat E rangle_n psi^*_nx t hat E psi_nx t textdx psi^*_nx t E_n psi_nx t textdx E_n psi^*_nx t psi_nx t textdx E_n abspsi_nx t^ textdx E_n where we have used the fact that the wavefunction is normalised. As expected the expectation value of an energy eigenstate corresponds to the energy level corresponding to that state. vspacemm The expectation value for langle hat E^ rangle_n can be derived in a similar way: langle E^ rangle_n psi^*_nx t hat E^ psi_nx t textdx psi^*_nx t hat E lefthat E psi_nx t right textdx psi^*_nx t hat E leftE_n psi_nx tright textdx E_n psi^*_nx t hat E psi_nx t textdx E_n psi^*_nx t E_n psi_nx t textdx E_n^ abspsi_nx t^ textdx E_n^ It follows for the standard deviation sigma sqrtlangle hat E^ rangle_n - leftlangle hat E rangle_nright^ sqrtE_n^ - E_n^ The energy of the energy eigenstate psi_n is exactly E_n. abc Applying the energy operator to the superposition state yields hat E psix t hat E leftfracsqrtpsi_x t + fracsqrtpsi_xtright fracsqrthat E psi_x t + fracsqrthat E psi_xt fracsqrtE_ psi_x t + fracsqrtE_ psi_xt It follows for the expectation value of the energy langle hat E rangle psi^*x t hat E psix t textdx leftfracsqrtpsi_^*x t + fracsqrtpsi_^*xtright leftfracsqrtE_ psi_x t + fracsqrtE_ psi_xt right textdx Because of the orthonormality of the eigenstates we know that psi^*_x t psi_x t textdx psi^*_x t psi_x t textdx The only relevant terms are thus fracsqrtpsi_^*x t fracsqrtE_ psi_x t textdx fracE_ psi_^*x t psi_x t textdx fracE_ and fracsqrtpsi_^*x t fracsqrtE_ psi_x t textdx fracE_ psi_^*x t psi_x t textdx fracE_ The expectation value is thus langle hat E rangle fracE_ + fracE_ fracE_+E_ Since the probabilities for the two eigenstates in the superposition are both equal to % this result is as expected. vspacemm In order to calculate the standard deviation we also have to determine the expectation value of the operator hat E^. This can be done in the same way as above but with the squares of the respective energy values. It follows that langle hat E^ rangle fracE_^ + fracE_^ fracE_^+E_^ The variance is therefore sigma^ langle hat E^ rangle - langle hat E rangle^ fracE_^+E_^ - leftfracE_+E_right^ frac E_^ + E_^ - fracE_^ + E_^ - E_ E_ fracE_^ + E_^ - E_ E_ fracE_ - E_^ For the standard deviation we find the following expression: sigma fracE_-E_ The standard deviation for the superposition of two energy eigenstates with the same probability of % corresponds to half the difference between the correpsonding energy values which is again quite uitive. abcliste