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NotesSet4 - PHY4221 Quantum Mechanics I Fall 2004 Examples...

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PHY4221 Quantum Mechanics I Fall 2004 Examples in One Dimension: 1. Free Particle The Hamiltonian operator ˆ H of a free particle consists of the kinetic energy term only: ˆ H = ˆ p 2 2 m . (1) Then what are the eigenvalues and eigenvectors of ˆ H ? Since any eigenvector of ˆ p is also an eigenvector of ˆ H , we can easily see that the eigenvalues of ˆ H are simply given by E p = p 2 2 m (2) where p ( -∞ , ) and the corresponding eigenvectors are {| p i} . It should be noted that there exists a two-fold degeneracy for each eigenvalue E p ; in other words, there are two different eigenvectors, namely {| ± p i} , associated with the same eigenvalue E p . Next, we consider the question: “How does the state vector of the free particle evolve in time?” To answer this question, we need to solve the time-dependent Schr¨ odinger equation: ˆ H | ψ ( t ) i = i ¯ h ∂t | ψ ( t ) i (3) whose solution is formally given by | ψ ( t ) i = exp - i t ¯ h ˆ H ± | ψ (0) i = Z -∞ dp | p ih p | exp - i t ¯ h ˆ H ± | ψ (0) i = Z -∞ dp | p i exp ( - i p 2 t 2 m ¯ h ) h p | ψ (0) i = ⇒ h p | ψ ( t ) i = exp ( - i p 2 t 2 m ¯ h ) h p | ψ (0) i . (4) 1
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If the initial state vector | ψ (0) i , or equivalently the initial wave function h p | ψ (0) i in the momentum space, of the free particle is given, then we can determine the subsequent time evolution of the system, namely determining the state vector | ψ ( t ) i . It is interesting to note that the wave function h p | ψ ( t ) i at any later time t > 0 differs from the initial wave function simply by a phase factor. Furthermore, the evolution operator exp
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NotesSet4 - PHY4221 Quantum Mechanics I Fall 2004 Examples...

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