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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2 You will notice that the evolution in the plane wave manner is actually quite general in the Schrödinger equation, as general as the separation of variables. We can also write the space equation, replacing X(x) with ϕ (x) as a more conventional symbol for wave function. ) ( ) ( ) ( ) ( 2 2 2 2 x E x x V x dx d m ψ φ = + h ( 4 . 7 ) h / ) ( ) , ( iEt e x t x = ( 4 . 8 ) ( Exercise 4.1 ) Is the plane wave for (x,t)=e i(kx- ω t) a necessary or sufficient condition for Eqs. (4.7) and (4.8) ? ( Exercise 4.2 ) For the procedure that we use to find the time evolution of free-particle (x,t) with (x,0)= 0 (x) through Fourier transforms by = dx e x ψ k A ikx ) ( ) ( 0 and = π dk e k A t x ψ t k ω kx i 2 ) ( ) , ( ) ) ( ( , what is different here by using h / ) ( ) , ( iEt e x t x = ? 4.2 Eigenfunctions of the Hamiltonian Equations (4.7) and (4.8) are called the time-independent Schrödinger equation. Although it is still difficult to solve, it is much easier to visualize the physical system: (x), as the eigenfunction for H ˆ , will evolve in time with h / ) ( iEt e x , where E is the corresponding eigenvalue. E is also the expectation value for the Hamiltonian operator H ˆ with the eigenfunction (x), since > =< = > < > < = > < > < >= < | ˆ | ) ( | ) ( ) ( | ˆ | ) ( ) , ( | ) , ( ) , ( | ˆ | ) , ( ˆ H E x x x H x t x t x t x H t x H (4.9) There is another property for the wave function h / ) ( ) , ( iEt e x t x = . Notice that ) ( ) ( ) ( ) ( ) , ( ) , ( ) , ( 2 * * x P x ψ x ψ x ψ t x ψ t x ψ t x P = = = = ( 4 . 1 0 ) The probability density P(x,t) does not have a time dependence for the eigenfunctions of H ˆ , and we thus refer the wave function h / ) ( ) , ( iEt e x t x = as a stationary state . Notice that this is ONLY true for each eigenfunction, but not for its linear superposition.
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note04 - ECE 3060: Introduction to Quantum and Statistical...

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