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Unformatted text preview: 1 EEE 352: Lecture 8 Electron Motion and Probability Velocity * Propagation of a wave packet ⇒ expansion of amplitude and phase ⇒ recognition of current ⇒ Classical analogy Ψ = Ψ + Ψ ∇ − E V m ) ( 2 2 2 r h Ernst Madelung David Bohm ( ) 2 2 1 2 1 2 1 2 2 2 1 2 4 k k k k A C k k k k T inc + = = = ψ ψ Why did we put this here? This is because both the reflection coefficient and the transmission coefficient are defined in terms of the probability current. 2 2 2 2 1 2 2 1 2 C m k C v J B m k B v J A m k A v J C d transmitte B reflected A incident h h h = = = = = = Last time, we introduced the idea of probability current. We need to pursue this a little further in order to garner the ideas of current in quantum mechanics. Now, we normally think of velocity as the description of the motion of a particle. Here, we are talking about waves. Previously, we connected the particle velocity to the group velocity of the wave, and used the momentum operator from which we can infer Hence, the momentum is an operator (it defines a differential operation on the wave function). ( ) ) ( ) ( t kx i t kx i ike e x ω ω − − = ∂ ∂ x i k p x ik ∂ ∂ − → = ∂ ∂ → h h In a similar manner, we could extract the position operator as and, once again, we find that this is a differential operator. We will see later that, since x and k are Fourier transform pairs, we can work in a position representation or a momentum representation, such that ( ) k i x ixe e k t kx i t kx i ∂ ∂ → = ∂ ∂ − − ) ( ) ( ω ω k k i momentum x i x position momentum position tion representa h h ∂ ∂ ∂ ∂ − ) , ( ) , ( t k t x ϕ ψ 2 By Fourier transform pairs, we mean that the two wave functions (one in the position representation and one in the momentum representation ) are related by a Fourier transform: ∫ ∫ ∞ ∞ − − ∞ ∞ − = = dx t x e t k dk t k e t x ikx ikx ) , ( 2 1 ) , ( ) , ( 2 1 ) , ( ψ π ϕ ϕ π ψ k k i momentum x i x position momentum position tion representa h h ∂ ∂ ∂ ∂ − ) , ( ) , ( t k t x ϕ ψ k k i momentum x i x position momentum position tion representa h h ∂ ∂ ∂ ∂ − ) , ( ) , ( t k t x ϕ ψ If we are in the position representation, then the momentum is an operator but, if we are in the momentum representation, then it is the position which is an operator x i k p ∂ ∂ − → = h h k i x ∂ ∂ = This now introduces an important point about the wave function. Let us look at the group velocity: x m i m k m p v g ∂ ∂ − → = = h h If the wave function is real, the velocity is imaginary! In order to have a real velocity, the wave function must be complex , and the velocity is related to the imaginary part of the wave function. Since the imaginary part is in the phase , the velocity is related to the phase of the wave function ....
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- Spring '08
- Uncertainty Principle, wave packet, wave function, K1, David Bohm