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BITTE GEBEN SIE KEINE ANTWORT, INDEM SIE WIE ANDERE KOPIEREN, SONST WERDE ICH BERICHTEN UND DOWNVOTEN 👎
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answerhappygod
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BITTE GEBEN SIE KEINE ANTWORT, INDEM SIE WIE ANDERE KOPIEREN, SONST WERDE ICH BERICHTEN UND DOWNVOTEN 👎
BITTE GEBEN SIE KEINE ANTWORT, INDEM SIE WIE ANDERE KOPIEREN, SONST WERDE ICH BERICHTEN UND DOWNVOTEN 
This is a sufficient condition for the existence of a bound state in a one-dimensional potential with finite range. It is not a necessary condition since, for instance, the one-dimensional harmonic oscillator is characterized by a potential energy that does not satisfy the inequality (10.81). In fact, it is well known that the one-dimensional harmonic oscillator has an infinite number of bound states (see Appendix B). **** Problem 10.6: Improvement of the criterion for existence of a bound state in one dimension This problem considers an improvement of the criterion (10.81) for the existence of a bound state in a one-dimensional potential V(2) with finite extent (i.e., assuming V(2) = 0 outside some interval [-a, +a]). Starting with a trial function of the form ox(2) = P(2) + XV (2), (10.82) where A is a real parameter and P(z) is the function defined as P(z) = 1 for 2 [-a, +a], P(z) = e for z> a, P(2)=e7 for z<-a, show that V(2) will have at least one bound state if ft V(z)dz ≤ 0 is satisfied. This is a weaker condition compared to the one derived in the previous problem, since even potential whose average value fa V(2)dz is equal to zero will have at least one bound state. (2+4)² (10.83) (10.84) (10.85)
This is a sufficient condition for the existence of a bound state in a one-dimensional potential with finite range. It is not a necessary condition since, for instance, the one-dimensional harmonic oscillator is characterized by a potential energy that does not satisfy the inequality (10.81). In fact, it is well known that the one-dimensional harmonic oscillator has an infinite number of bound states (see Appendix B). **** Problem 10.6: Improvement of the criterion for existence of a bound state in one dimension This problem considers an improvement of the criterion (10.81) for the existence of a bound state in a one-dimensional potential V(2) with finite extent (i.e., assuming V(2) = 0 outside some interval [-a, +a]). Starting with a trial function of the form ox(2) = P(2) + XV (2), (10.82) where A is a real parameter and P(z) is the function defined as P(z) = 1 for 2 [-a, +a], P(z) = e for z> a, P(2)=e7 for z<-a, show that V(2) will have at least one bound state if ft V(z)dz ≤ 0 is satisfied. This is a weaker condition compared to the one derived in the previous problem, since even potential whose average value fa V(2)dz is equal to zero will have at least one bound state. (2+4)² (10.83) (10.84) (10.85)