note02 - Introduction to Quantum and Statistical Mechanics...

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1 Introduction to Quantum and Statistical Mechanics Fall 2009 Handout 2 The Schrödinger Equation from Intuition and as a Postulate Complementary Reading: Chap. 4 (pp. 43 – 64), Hagelstein, Senturia and Orlando 2.1 Wave packet for intuitive construction of the Schrödinger equation After learning the central concept of deBroglie wavelength, we can intuitively understand how classical particle mechanics and wave mechanics can be unified in the eigenfunction of the wave equation as a “wave packet” (or similarly as superposition of plane waves) by a(x)e i(kx- ω t) , where () ψ h h h h = = = = E k p t E x p i t kx i t x exp ) ( exp ) , ( (2.1) The question in hand is: what is the overarching equation that can capture this wave-particle duality and satisfy the experimental evidence from photons and electrons. Namely, k p h = and h = E . In the wave equation, if we substitute the plane wave into Eq. (1.5), we get a dispersion relation of k c ω ± = , where the wave velocity c is also the group velocity k ω v g = , and the phase velocity v p . We will now see what dispersion relation and governing equation can be constructed for particle-like behavior. The classical kinetic energy of a free particle is: m k k ω ω m p E 2 ) ( 2 2 2 h h = = = (2.2) We now ask the question: what relation of the derivatives in time and space will give this quadratic dispersion relation? For the plane wave, we know: ) ( ) ( ) ( ) ( ; t ω kx i t ω kx i t ω kx i t ω kx i ike e x e ω i e t = = , and the dispersion relation of m k k ω 2 ) ( 2 h = can be obtained from: ) , ( 2 ) , ( 2 2 2 t x ψ x m t x ψ t i = h h or in 3D: ) , ( 2 ) , ( 2 2 t x m t x t i v h v h = (2.3) Remember that the dispersion relation is obtained from equating the wave energy ω E h = with the free-particle kinetic energy E = p 2 /2m . This suggests that if we map the two “operators” on the plane wave function: x i p t i E = = h h ; or in 3D: i i x i p = h or = h v i p (2.4)
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2 We can then obtain the dispersion relation of Eq. (2.2) and the wave behavior of a particle! ( Exercise 2.1 ) What is the group velocity ( v g ) behind the dispersion relation of m k k ω 2 / ) ( 2 h = ? What does it mean when v g depends on the wave vector k or the deBroglie wavelength? Equation (2.4) is however rather strange when the imaginary number and the operator were introduced. This is counter-intuitive as the real world is described by imaginary number and operators! However, this will eventually become the most fundamental description of the world we understand today. We will now proceed to obtain a consistent mathematical formulation to unify the classical and wave mechanics in the spirit of the deBroglie particle-wave duality. The above description is only for free particles. If we add the potential energy V(x) (or in 3D, ) ( x V v ), which depends ONLY on x but not on p , we will have the Hamilton (K.E. + potential), in the same meaning in the classical mechanics, but with a modified definition of momentum and energy, as: ) , ( ) ( ) , ( 2 ) , ( 2 2 2 t x x V t x x m t x H
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This note was uploaded on 11/26/2010 for the course ECE 3060 at Cornell University (Engineering School).

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note02 - Introduction to Quantum and Statistical Mechanics...

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