Chapter6_30 - lowering operator to the lowest energy state...

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

View Full Document Right Arrow Icon
PHY3063 R. D. Field Department of Physics Chapter6_30.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 | ) ( | ) )( ( | ]) ) ( , [ ) (( | | ) ( h h Thus, > + >= + h n n n op E c E a | | ) ( where c n are constants (that may depend on n ) and similarly > >= 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 h . Thus, 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 h 2 1 0 E . The state |E 0 > is the state of lowest energy and E 0 is the ground state energy. Applying the
Background 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 &gt; &gt;= h | | ) ( E d E a op . But there is no state with energy lower than E which implies that d = 0 and | ) ( &gt;= E a op . Now we can solve for E as follows &gt; &gt;= &gt;= + &gt;= + 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 | &gt;= &lt; E E . Excited States: All the other states are calculated from the ground state using the raising operator as follows: &gt; &gt;= + | ) ( ! 1 | E a n E n op n and &gt; + &gt;= n n op E n E H | ) ( | 2 1 h and hf n n E n ) ( ) ( 2 1 2 1 + = + = h Plancks guess was E n = nhf!...
View Full Document

Ask a homework question - tutors are online