Superposition State
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Exercise:
For the infinite well the system is in a non-stationary state given by the superposition abcliste abc psixt sqrtfracpsi_xt+sqrtfracpsi_xt abc psixt fracsqrtpsi_xt+fracsqrtpsi_xt abcliste Calculate the respective expectation value and the uncertay for the energy of the system.
Solution:
abcliste abc The expectation value is langle E rangle fracE_+fracE_ fracpi^hbar^mL^left^+ ^right fracpi^hbar^mL^ left fracE_ approx EaP times E_ right The variance is sigma^ langle E^ rangle - langle E rangle^ fracE_^+fracE_^-leftfracE_+fracE_right^ fracE_^+fracE_^-fracE_^-fracE_^-fracE_E_ fracE_^+fracE_^-fracE_E_ fracleftE_-E_right^ It follows for the uncertay standard deviation sigma fracsqrt leftE_-E_right fracsqrt pi^hbar^mL^left^-^right fracsqrtpi^hbar^mL^ left fracsqrt E_ approx siaP times E_right abc The expectation value is langle E rangle fracE_+fracE_ fracpi^hbar^mL^left^+ ^right fracpi^hbar^mL^ left fracE_ approx EbP times E_ right The variance is sigma^ langle E^ rangle - langle E rangle^ fracE_^+fracE_^-leftfracE_+fracE_right^ fracE_^+fracE_^-fracE_^-fracE_^-fracE_ E_ fracE_^+fracE_^-fracE_ E_ fracleftE_-E_right^ It follows for the uncertay standard deviation sigma fracleftE_-E_right fracpi^hbar^mL^left^-^right fracpi^hbar^mL^ left fracE_ approx sibPtimes E_right abcliste The figure below displays the first three energy levels black lines the expectation value red line and the uncertay shaded red band for the situations in a left and b right. center includegraphicswidthcm#image_path:expectation-valuand-uncertay-# center
For the infinite well the system is in a non-stationary state given by the superposition abcliste abc psixt sqrtfracpsi_xt+sqrtfracpsi_xt abc psixt fracsqrtpsi_xt+fracsqrtpsi_xt abcliste Calculate the respective expectation value and the uncertay for the energy of the system.
Solution:
abcliste abc The expectation value is langle E rangle fracE_+fracE_ fracpi^hbar^mL^left^+ ^right fracpi^hbar^mL^ left fracE_ approx EaP times E_ right The variance is sigma^ langle E^ rangle - langle E rangle^ fracE_^+fracE_^-leftfracE_+fracE_right^ fracE_^+fracE_^-fracE_^-fracE_^-fracE_E_ fracE_^+fracE_^-fracE_E_ fracleftE_-E_right^ It follows for the uncertay standard deviation sigma fracsqrt leftE_-E_right fracsqrt pi^hbar^mL^left^-^right fracsqrtpi^hbar^mL^ left fracsqrt E_ approx siaP times E_right abc The expectation value is langle E rangle fracE_+fracE_ fracpi^hbar^mL^left^+ ^right fracpi^hbar^mL^ left fracE_ approx EbP times E_ right The variance is sigma^ langle E^ rangle - langle E rangle^ fracE_^+fracE_^-leftfracE_+fracE_right^ fracE_^+fracE_^-fracE_^-fracE_^-fracE_ E_ fracE_^+fracE_^-fracE_ E_ fracleftE_-E_right^ It follows for the uncertay standard deviation sigma fracleftE_-E_right fracpi^hbar^mL^left^-^right fracpi^hbar^mL^ left fracE_ approx sibPtimes E_right abcliste The figure below displays the first three energy levels black lines the expectation value red line and the uncertay shaded red band for the situations in a left and b right. center includegraphicswidthcm#image_path:expectation-valuand-uncertay-# center