Lecture_Notes_3

Lecture_Notes_3 - PHYS4221 Quantum Mechanics I Fall 2010 2...

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Unformatted text preview: PHYS4221 Quantum Mechanics I Fall 2010 2. 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) 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 1 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 n- i t ¯ h ˆ H o of the system can be represented by exp- i t ¯ h ˆ H = Z ∞-∞ dp | p i exp (- i p 2 t 2 m ¯ h ) h p | . Similarly, if the initial wave function h x | ψ (0) i in the coordinate space is given instead, then the state vector | ψ ( t ) i can be formally expressed as | ψ ( t ) i = Z ∞-∞ dx | x ih x | exp- i t ¯ h ˆ H | ψ (0) i = Z ∞-∞ dx | x i exp ( i t ¯ h 2 m ∂ 2 ∂x 2 ) h x | ψ (0) i = Z ∞-∞ dx | x ih x | exp- i t ¯ h ˆ H Z ∞-∞ dx | x ih x | | ψ (0) i = Z ∞-∞ dx | x i Z ∞-∞ dx K ( x,t ; x , 0) h x | ψ (0) i = ⇒ h x | ψ ( t ) i = Z ∞-∞ dx K ( x,t ; x , 0) h x | ψ (0) i . (5) where the kernel K ( x,t ; x , 0) ≡ h x | exp n- i t ¯ h ˆ H o | x i is given by K ( x,t ; x , 0) = h x | exp- i t ¯ h ˆ H Z ∞-∞ dp | p ih p | | x i = Z ∞-∞ dp exp (- i p 2 t 2 m ¯ h ) h x | p ih p | x i = Z ∞-∞ dp 2 π ¯ h exp (- i p 2 t 2 m ¯ h ) exp ( i p ( x- x ) ¯ h ) = r m 2 πi ¯ ht exp- m 2 i ¯ ht ( x- x ) 2 . (6) The kernel K ( x,t ; x , 0) is also commonly known as the propagator of the free particle....
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Lecture_Notes_3 - PHYS4221 Quantum Mechanics I Fall 2010 2...

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