Orthonormal wave functions
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Exercise:
Show that the wave functions for the eigenstates of the infinite potential well psi_nx t sqrtfracL sink_n x are orthonormal i.e. they fulfil the relation _^L psi^*_mxtpsi_nxttextrmdx delta_mn where delta_mn is the Kronecker symbol: delta_mn cases & textrmif mn & textrmelse cases
Solution:
We first consider the case mn. _^L|psi_nxt|^textrmdx fracL _^Lsin^k_n x textrmdx fracL fracL where we have used the identity sin^alpha frac left-cos alpharight to evaluate the egral _^L sin^k_n x textrmdx _^L frac left - cosk_n xright textrmdx fracL - frack_nleftsink_n xright_^L fracL - frack_nleft-right fracL In the last step we have used that sin k_n L sinleft fracnpiL L right sin n pi For the case mneq n we have to evaluate the egral I_mn _^L psi_nxt psi_m^*xt textrmdx fracL _^L sink_n x sink_m x textrmdx With the identity sinalpha sinbeta frac leftcosalpha-beta - cosalpha+betaright this can be written as I_mn fracL _^L leftcosleftk_n-k_m xright - cosleftk_n+k_m xright right textrmdx fracL leftfrack_n-k_m sinleftk_n-k_m xright - frack_n+k_m sinleftk_n+k_m xright right_^L At x and xL all sine terms vanish e.g. sinleftk_n-k_m Lright sinleftn-mpiright This leads to the expected result: I_mn
Show that the wave functions for the eigenstates of the infinite potential well psi_nx t sqrtfracL sink_n x are orthonormal i.e. they fulfil the relation _^L psi^*_mxtpsi_nxttextrmdx delta_mn where delta_mn is the Kronecker symbol: delta_mn cases & textrmif mn & textrmelse cases
Solution:
We first consider the case mn. _^L|psi_nxt|^textrmdx fracL _^Lsin^k_n x textrmdx fracL fracL where we have used the identity sin^alpha frac left-cos alpharight to evaluate the egral _^L sin^k_n x textrmdx _^L frac left - cosk_n xright textrmdx fracL - frack_nleftsink_n xright_^L fracL - frack_nleft-right fracL In the last step we have used that sin k_n L sinleft fracnpiL L right sin n pi For the case mneq n we have to evaluate the egral I_mn _^L psi_nxt psi_m^*xt textrmdx fracL _^L sink_n x sink_m x textrmdx With the identity sinalpha sinbeta frac leftcosalpha-beta - cosalpha+betaright this can be written as I_mn fracL _^L leftcosleftk_n-k_m xright - cosleftk_n+k_m xright right textrmdx fracL leftfrack_n-k_m sinleftk_n-k_m xright - frack_n+k_m sinleftk_n+k_m xright right_^L At x and xL all sine terms vanish e.g. sinleftk_n-k_m Lright sinleftn-mpiright This leads to the expected result: I_mn