Exercise:
The position operator hat x corresponding to the position x is defined as a multiplication by x. It follows that the expectation value for the position of a quantum mechanical particle e.g. an electron in a state described by the wave function psixt is given by langle x rangle psi^*xt x psixt textrmdx |psixt|^ x textrmdx abcliste abc Show that the expectation value for a particle in a stationary energy eigenstate has the value langle x rangleL/ as expected. abc Derive the uncertay standard deviation for a position measurement if the system is in an energy eigenstate. abcliste

Solution:
abcliste abc The expectation value for the nth energy level is given by langle x rangle A^_^L sin^k_n x x textrmdx fracA^_^Lleft-cosk_n xright x textrmdx fracA^_^L x textrmdx - fracA^ _^Lcos k_n x x textrmdx where we have used the identy sin^alpha fracleft-cosalpharight The first egral can easily be evaluated: _^L x textrmdx fracx^ Big|_^L fracL^ fracL The second egral can be solved through egration by parts with fx x and g'x cosk_n x: _^Lcos k_n x x textrmdx fracsin k_n x k_n x Big|_^L - _^L fracsin k_n x k_n textrmdx fracsin k_n L k_n L + fraccos k_n x k_n^ Big|_^L fracsinpi n k_n L + fraccospi n-cos k_n^ + frac- k_n^ It follows for the expectation value langle x rangle fracA^ fracL^ fracfracLfracL^ fracL abc The uncertay is sigma sqrtlangle x^ rangle - langle x rangle^ We already know that langle x rangle fracL The expectation value for x^ is langle x^ rangle A^_^L sink_n x x^ textrmdx fracA^_^L left-cosk_n xright x^ textrmdx The first term in the egral yields _^L x^ textrmdx fracx^Big|_^L fracL^ For the second term we use egration by parts with fx x^ and g'x cosk_n x: _^L cos k_n x x^ textrmdx fracsin k_n x k_n x^ Big|_^L - _^L fracsin k_n x k_nx textrmdx fracsinpi n-sink_n-_^L fracsin k_n x k_nx textrmdx - _^L fracsin k_n x k_n x textrmdx We use egration by parts again with fx x and g'x sink_n x: dots - leftfrac-cos k_n x k_n^ x Big|_^L - _^L frac-cos k_n x k_n textrmdx right fraccospi nL k_n^+_^L fraccos k_n xk_n textrmdx fracLpi^/L^ + fracsin k_n x k_n^ Big|_^L fracL^ n^ pi^ + fracsinpi-sink_n^ fracL^ n^ pi^ Collecting all the terms we find langle x^ rangle fracA^ leftfracL^ - fracL^ n^ pi^ right fracL L^ leftfrac-frac n^ pi^ right L^ leftfrac-frac n^ pi^right The variance sigma^ is thus sigma^ langle x^ rangle - langle x rangle ^ L^ leftfrac-frac n^ pi^ - frac right L^ leftfrac - frac n^ pi^ right L^ fracn^ pi^ - n^ pi^ and the uncertay sigma sqrtlangle x^ rangle - langle x rangle ^ L sqrtfracn^ pi^ - n^ pi^ For n the relative uncertay is fracsigmaL siP For high quantum numbers ntoinfty the relative uncertay ts to fracsigma_inftyL siinfF siinfP The energy eigenstates are no eigenvectors of the position operator! abcliste