Momentum of Stationary States
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Exercise:
For the infinite potential well show that the expectation value for the momentum of a stationary state is equal to zero and that the uncertay is sigma_p fracnpi hbarL
Solution:
The expectation value for the stationary state n is langle p rangle_n _^L psi_n^*xt hat p psi_nxt textrmdx _^L psi_n^*xt fractextrmdtextrmdxpsi_nxt textrmdx _^L A^* sink_n x -ihbarfractextrmdtextrmdxAsink_n x textrmdx |A|^_^L sink_n xcosk_n x k_n textrmd x k_n |A|^ _^L sink_n xcosk_n x textrmdx With the trigonometric identity sinalphacosbeta fracleftsinalpha-beta+sinalpha+betaright the egrand can be written as sinkn xcosk_n x fracleftsin+sin k_n xright fracsin k_n x The egral of sin k_n x is _^L sin k_n x textrmdx -fraccos k_n x k_n Big|_^L -fraccos k_n L-cos k_n -fraccos pi n-cos k_n -frac - k_n This proves the first statement. vspacemm The expectation value for p^ is langle p^ rangle_n _^L psi_n^*xt hat p^ psi_nxt textrmdx _^L psi_n^*xt -hbar^fractextrmd^textrmdx^A sink_n xe^iomega t textrmdx -hbar^_^L psi_n^*xt-Asink_n xk_n^ e^iomega t textrmdx hbar^ k^ _^L psi_n^*xt psi_nxt textrmdx hbar k_n^ where we have used the normalisation condition for the state psi_n. vspacemm It follows for the uncertay sigma_pn sqrtlangle p^ rangle_n - langle p rangle_n^ sqrthbar k_n^ - hbar k_n hbar fracnpiL fracnpihbarL
For the infinite potential well show that the expectation value for the momentum of a stationary state is equal to zero and that the uncertay is sigma_p fracnpi hbarL
Solution:
The expectation value for the stationary state n is langle p rangle_n _^L psi_n^*xt hat p psi_nxt textrmdx _^L psi_n^*xt fractextrmdtextrmdxpsi_nxt textrmdx _^L A^* sink_n x -ihbarfractextrmdtextrmdxAsink_n x textrmdx |A|^_^L sink_n xcosk_n x k_n textrmd x k_n |A|^ _^L sink_n xcosk_n x textrmdx With the trigonometric identity sinalphacosbeta fracleftsinalpha-beta+sinalpha+betaright the egrand can be written as sinkn xcosk_n x fracleftsin+sin k_n xright fracsin k_n x The egral of sin k_n x is _^L sin k_n x textrmdx -fraccos k_n x k_n Big|_^L -fraccos k_n L-cos k_n -fraccos pi n-cos k_n -frac - k_n This proves the first statement. vspacemm The expectation value for p^ is langle p^ rangle_n _^L psi_n^*xt hat p^ psi_nxt textrmdx _^L psi_n^*xt -hbar^fractextrmd^textrmdx^A sink_n xe^iomega t textrmdx -hbar^_^L psi_n^*xt-Asink_n xk_n^ e^iomega t textrmdx hbar^ k^ _^L psi_n^*xt psi_nxt textrmdx hbar k_n^ where we have used the normalisation condition for the state psi_n. vspacemm It follows for the uncertay sigma_pn sqrtlangle p^ rangle_n - langle p rangle_n^ sqrthbar k_n^ - hbar k_n hbar fracnpiL fracnpihbarL