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NotesSet5 - PHY4221 Quantum Mechanics I Fall 2004 More...

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PHY4221 Quantum Mechanics I Fall 2004 More Examples in One Dimension: 1. Particle in a Delta-function Potential Well The Hamiltonian operator ˆ H of a particle in a Dirac delta-function potential well is given by ˆ H = ˆ p 2 2 m - V 0 δ x ) . (1) To ﬁnd the eigenvalues and eigenvectors of the bound states ( i.e. those eigenstates with negative eigenvalues) of ˆ H , we try to solve the eigenvalue equation in the momentum space: h p | ˆ H | Φ i = h p | " ˆ p 2 2 m - V 0 δ x ) # | Φ i = E h p | Φ i , for E < 0 = p 2 2 m h p | Φ i - h p | •Z -∞ dx | x ih x | V 0 δ x ) | Φ i = E h p | Φ i = p 2 2 m h p | Φ i - V 0 Z -∞ dx h p | x i δ ( x ) h x | Φ i = E h p | Φ i = p 2 2 m h p | Φ i - V 0 2 π ¯ h h x = 0 | Φ i = E h p | Φ i = ⇒ h p | Φ i = 2 mV 0 2 π ¯ h ( p 2 - 2 mE ) h x = 0 | Φ i = ⇒ h x | Φ i = h x | •Z -∞ dp | p ih p | | Φ i = mV 0 π ¯ h Z -∞ dp exp n i px ¯ h o p 2 + 2 m | E | h x = 0 | Φ i . (2) Exercise: Show that Z -∞ dx exp {- ikx } exp {- κ | x |} = 2 κ κ 2 + k 2 . With the result shown in the above exercise, we could simplify Eq.(2) as h x | Φ i = s m 2 | E | ± V 0 ¯ h exp - q 2 m | E | ¯ h | x | h x = 0 | Φ i . (3) 1

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Setting x = 0 on both sides of Eq.(3) yields h x = 0 | Φ i = s m 2 | E | ± V 0 ¯ h h x = 0 | Φ i = E = - mV 2 0 h 2 . (4) Accordingly, there exists one and only one bound state of the system. The correspond- ing eigenfunction (in both coordinate space and momentum space) can be obtained as
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NotesSet5 - PHY4221 Quantum Mechanics I Fall 2004 More...

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