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note04 - ECE 3060 Introduction to Quantum and Statistical...

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1 ECE 3060: Introduction to Quantum and Statistical Mechanics Handout 4 Time-Independent Schrödinger Equation and Time Evolution Textbook Reading: Hagelstein, Senturia and Orlando, Chap. 6 (pp. 89-110). 4.1. Separation of space and time dependence in Schrödinger equation By applying the principle of the eigenvalue problems to the Schrödinger equation, we can come up with an important subclass of quantum mechanics when the potential V(x) does not change with time. We will examine the “time-dependent” Schrödinger equation: ( ) ( ) ( ) ( ) ( ) t x ψ t i t x ψ E t x ψ x V t x ψ x m t x ψ H , , ˆ , ) ( , 2 , ˆ 2 2 2 = = + = h h (4.1) We will use the principle of separation of variables and assume that the wave function can be decoupled as: ) ( ) ( ) , ( t T x X t x ψ = (4.2) Notice that this will just include a sub-class of wave functions, as there are many functions where the separation of space and time is not possible. Substituting X(x)T(t) into the Schrödinger equation and doing some term rearrangement, we obtain: E x X x X x V x X dx d m t T t T dt d i = + = ) ( ) ( ) ( ) ( 2 ) ( ) ( 2 2 2 h h (4.3) The first term depends ONLY on t , while the second term depends ONLY on x . Therefore, they must be equal to a constant, here we name it, E (it will be the energy eigenvalue indeed). We will now decouple the Schrödinger equation to two equations that need to be simultaneously satisfied: ) ( ) ( t ET t T dt d i = h (4.4) ) ( ) ( ) ( ) ( 2 2 2 2 x EX x X x V x X dx d m = + h (4.5) The time equation above is easy to solve, and we have h / ) ( iEt Ce t T = (4.6)
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