{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Chapter2_14 - “lowering operator” to the lowest energy...

Info icon This preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
PHY4604 R. D. Field Department of Physics Chapter2_14.doc University of Florida The Harmonic Oscillator (3) Energy Eigenvalue Equation: We are looking for solutions of the equation H op |E n > = E n |E n > where |E n > are the “eigenkets” and E n are the allowed energies ( i.e. eigenvalues). The state (a + ) op |E n > = |a + E n > is an “eigenket” with energy ω h + n E since > + >= + = > + >= >= + + + + + + n n n op n n op op op op n op n op op E a E E a E E a H H a E a H E a H | ) ( | ) )( ( | ]) ) ( , [ ) (( | | ) ( ω ω h h Thus, > + >= + ω h n n n op E c E a | | ) ( where c n are constants (that may depend on n ) and similarly > >= ω h n n n op E d E a | | ) ( Ground State Energy (Lowest Energy State): We know that the norm of the state (a - ) op |E n > = | a - E n > must be positive definite and hence > >=< >=< ≤< + n op n n op op n n n E H E E a a E E a E a | | | ) ( ) ( | | 0 2 1 1 ω h . Thus, ω h 2 1 | | >≥ < n op n E H E . This implies that there is a minimum energy state which we will call |E 0 > with H op |E 0 > = E 0 |E 0 > where ω h 2 1 0 E . The state |E 0 > is the state of lowest energy and E 0 is the ground state energy.
Image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: “lowering operator” to the lowest energy state gives > − >= − h | | ) ( E d E a op . But there is no state with energy lower than E which implies that d = 0 and | ) ( >= − E a op . Now we can solve for E as follows > >= >= + >= − + 2 1 2 1 | | | ) ) ( ) ( ( | E E E E a a E H op op op h h h . Hence, h 2 1 = E and we normalize so that 1 | >= < E E . Excited States: All the other states are calculated from the ground state using the “raising operator” as follows: > >= + | ) ( ! 1 | E a n E n op n and > + >= n n op E n E H | ) ( | 2 1 h and hf n n E n ) ( ) ( 2 1 2 1 + = + = h Planck’s guess was E n = nhf!...
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern