c120a_lecture14 - Chem 120A Spring 2007 Particle-in-a-box...

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Chem 120A Particle-in-a-box 02/16/07 Spring 2007 Lecture 14 READING for this week: Ratner and Schatz, Chapter 3; also recommended: Atkins and Friedman, Chapters 1, 2.1-2.4, 2.12-2.13; Feynman Lectures in Physics vol III, Chapter 8-1 to 8-6; 1 Recap: time dependent and time independent Schrodinger equations We first review the relation between the time dependent and time independent forms of the Schrodinger equation. The former is usually referred to simply as the Schrodinger equation, while the latter is often called the energy eigenvalue equation. The Schrodinger equation i ¯ h t ψ ( x , t ) = ˆ H ψ ( x , t ) (1) defines the relationship between the energy of a system ( through ˆ H ) and its time development. Can we understand this relationship better? First, we note that this is a partial differential equation, which means that it is a differential equation with more than one variable (x and t in this case). We now employ a handy math trick for partial differential equations, and assume that the solution to the Schrodinger Equation can be written as the product ψ ( x , t ) = ψ ( x ) φ ( t ) . This is called separation of variables . If we plug this into the Schr. Eqn. and divide both sides by ψ ( x ) φ ( t ) : i ¯ h ψ ( x ) t φ ( t ) = φ ( t ) ˆ H ψ ( x ) i ¯ h ∂φ t φ ( t ) = ˆ H ψ ( x ) ψ ( x ) (2) The left hand side (LHS) is a function of t, and the right hand side (RHS) is a function of x. Therefore, for this solution to make sense for all possible x and t, both sides must equal a constant . What is that constant? Why, energy , of course! So, the Schr. eqn. breaks into two equations, one in time (t) and one in space (x): space : ˆ H ψ ( x ) = E ψ ( x ) (3) time : i ¯ h ∂φ ( t ) t = E φ ( t ) (4) Chem 120A, Spring 2007, Lecture 14 1
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This time equation is easy: φ ( t ) = e - iEt / ¯ h φ ( t ) = e - i ω t , ω = E ¯ h , or E = ¯ h ω . (Planck-Einstein relation.) The spatial equation is harder. It is called the ”time-independent Schrodinger equation.” However difficult to solve, this equation is a special type of equation known as an eigenvalue problem, where ”operator × ψ = ”constant × ψ .” Thus the Schr. eqn. reduces to an eigenvalue problem, which has a storied history in mathematics. To solve the Schr. eqn., one must find the set of wavefunctions { ψ k ( x ) } that return to themselves (times a constant) when acted on by ˆ H : ˆ H ψ k ( x ) = E k ψ k ( x ) . Each solution ψ k has an energy E k associated with it. The states { ψ k } and only they have well-defined energies. The energies { E k } are the set of all physically allowed energies of the system. The full time-dependent solution is ψ ( x , t ) = ψ ( x ) φ ( t ) = ψ k ( x ) e - iE k t / ¯ h . Since these are linear equations we can add solutions, just as above for the free particle case. So, the most general solution to the Schrodinger equation is ψ ( x , t ) = k A k ψ k ( x ) e - iE
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This note was uploaded on 02/19/2010 for the course CHEM 120A taught by Professor Whaley during the Spring '07 term at Berkeley.

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c120a_lecture14 - Chem 120A Spring 2007 Particle-in-a-box...

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