Ch2 - SJP QM 3220 Ch. 2, part 1 Once again, the Schrdinger...

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SJP QM 3220 Ch. 2, part 1 Page 1 (S. Pollock, taken from M. Dubson) with thanks to J. Anderson for typesetting. Fall 2008 Once again, the Schrödinger equation: (which can also be written Ĥ ψ (x,t) if you like.) And once again, assume V = V(x) (no t in there!) We can start to solve the PDE by SEPARATION OF VARIABLES . Assume (hope? wonder if?) we might find a solution of form Ψ (x,t) = u(x) φ (t). Griffiths calls u(x) ψ (x), but I can't distinguish a "small ψ " from the "capital Ψ " so easily in my handwriting. You’ll find different authors use both of these notations…) So Note full derivative on right hand side! So Schrödinger equation reads (with d φ dt , and du dx u ' ) Now divide both sides by Ψ = u • φ . i ( t ) ( t ) function of time only = 2 2 m u ''( x ) u ( x ) + V ( x ) function of space only This is not possible unless both sides are constants . Convince yourself; that is the key to the "method of separation of variables". Let's name this constant "E". [Note units of E are or simply V(x), either way, check, it's Energy !]
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SJP QM 3220 Ch. 2, part 1 Page 2 (S. Pollock, taken from M. Dubson) with thanks to J. Anderson for typesetting. Fall 2008 So (1) (2) These are ordinary O.D.E.'s Equation (1) is about as simple as ODE's get! Check any constant, it's a linear ODE. (1 st order linear ODE is supposed to give one undetermined constant, right?) This is "universal", no matter what V (x) is, once we find a u(x), we'll have a corresponding But be careful, that u(x) depends on E, (2) . this is This is the "time independent Schrödinger equation". You can also write this as which is an "eigenvalue equation". Ĥ = "Hamiltonian" operator = In general, has many possible solutions. eigenfunctions eigenvalues u 1 (x), u 2 (x), … u n (x) may all work, each corresponding to some particular eigenvalue E 1 , E 2 , , E n . (What we will find is not any old E is possible if you want u(x) to be normalizable and well behaved. Only certain E's, the E n 's, will be ok.) Such a u n (x) is called a "stationary state of Ĥ ". Why? Let’s see…
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SJP QM 3220 Ch. 2, part 1 Page 3 (S. Pollock, taken from M. Dubson) with thanks to J. Anderson for typesetting. Fall 2008 Notice that Ψ n (x,t) corresponding to u n is ◄══ go back a page! So ◄══ no time dependence (for the probability density ). It's not evolving in time; it's "stationary". (Because ) Convince yourself! (If you think back to de Broglie's free particle with E = , it looks like we had stationary states Ψ n (x,t) with this (same) simple time dependence, e i ω t , with . This will turn out to be quite general) If you compute ˆ Q (the expectation value of any operator “Q” for a stationary state) the e iE n t / in Ψ multiplies the e iE n t / in Ψ * , and goes away… (This is assuming the operator Q depends on x or p, but not time explicitly) again, no time dependence.
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Ch2 - SJP QM 3220 Ch. 2, part 1 Once again, the Schrdinger...

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