df(x) dx = h(x) Chemistry 4531 Mathematical Preliminaries Spring 2009 I. A Primer on Differential Equations Order of differential equation

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1 Chemistry 4531 Mathematical Preliminaries Spring 009 I. A Primer on Differential Equations Order of differential equation Linearity of differential equation Partial vs. Ordinary Differential Equations Quantum Mechanics involves primarily nd order linear partial differential equations First order linear ordinary df(x) = h(x) one arbitrary constant 10

2 Second order linear ordinary d f(x) +a f(x) = 0 There must be two arbitrary constants in the general solution. The solution is ( ) ( ) f( x) = Asin ax + Bcos ax or ( ) f( x) = Csin ax+ D or ( ) f( x) = Ecos ax+ F or even f(x) = Ge i ( ax) i + He ( ax) The values of the constants are determined by boundary conditions on the differential equations. How do we know which form to use?? It cannot matter, so we use the one that is most convenient. 11

3 Taylor Series expansion of a function f(x) about x = x 0 df(x) d f(x) (x x 0 ) f(x) f(x 0) + (x x 0) ! x=x0 x=x0 Convergence Radius of Convergence Absolute convergence expand e ix about x =0 e ix ix i0 = e + de x=0 ix x + de x=0 x 3 ix 3! + de x=0 3 x 3! +... sin(x) = cos(x) = SO ix e = cos(x) + i sin (x) The EULER FORMULA What is the magnitude of e ix? 1

4 Odd and Even Functions Odd: F(x) = -F(-x) Even: F(x) = F(-x) This definition refers to the symmetry behavior (parity) about the origin, x=0. The definition can be readily generalized to describe symmetry properties about any point, say x = a. e.g. Odd: F(x-a) = -F(-[x-a]) Even: F(x-a) = F(-[x-a]) Using Mathcad, let us look at examples and also some integrals. x 10, F( x ) ( x ) The function is even about x=. It is neither odd nor even about any other point. Let's look at an odd function about x=: Gx ( ) = 64 sin( x ) 3 64 [ ] What can we say about integrals centered at x=? Let us evaluate some integrals involving F and G. We do so by inspection!! 13

5 4 4 F( x ) F( x ) Gx ( ) Gx ( ) F( xgx ) ( ) F( xgx ) ( ) OPERATORS An operator is an instruction telling you what to do with a function, much like a function is an instruction telling you what to do with a number. Examples of operators include d/, *,,, etc. We will denote a general symbolic operator with a "hat" over the symbol, e.g. Â So if Â =, then Â f(x) = f( x) An entire operator algebra can be derived. Notation: ÂĈf(x) Â(Ĉf(x)) Ĉ f(x) Ĉ [Ĉf(x)] (Â + Ĉ) f(x) Â (x) + Ĉ f(x) The ORDER of operators can make a difference! A very important concept! Â Ĉ f(x)? = Ĉ Â f(x) DEFINITION: Â Ĉ - Ĉ Â [Â, Ĉ], the COMMUTATOR of Â and Ĉ. If the commutator is zero for any function f, then the operators are said to commute. Namely, the order of carrying out the operations Â and Ĉ does not matter. Operators and commutation are critical concepts in quantum mechanics and are directly related to measurement. Example: let Â = x and Ĉ = d/. All Quantum Mechanical Operators are linear. Definition: Â is linear if and only if 14

6 Â (c 1 f + c g) = c 1 Â f + c Â g where c 1 and c are constants and f and g are any two functions. Which of these operators is linear?? R = ( ) ( ) d P = x d( ) 3 3 d( ) ˆM = x The operator equation Â(f(x)) = c f(x), where c is a constant, is called an eigenvalue equation. The function f(x) is called the eigenfunction of the operator Â, with eigenvalue c. It turns out that in quantum mechanics all physical properties are represented by operators and that the possible values of that property are related to the eigenvalues of that operator. Thus much of our work will be related to finding eigenvalues and eigenfunctions of operators. Examples: I. Let Â= d/. Then we must find f(x) such that df = c f(x) The solution in this case is f(x) = e cx, and the eigenvalue is c (where c is any number) II. Find the eigenvalues and eigenfunctions of Â = -d /. That is, find all a i and f i (x) such that dfi( x) = af i i( x). Rewriting, 15

7 d f i (x) aif i(x) = 0 a standard nd order homogeneous differential equation with solution: f i(x) = Asin( ai x ) +Bcos( ai x ), where a i is real, and A and B are any numbers. POTENTIAL ENERGY We will consider the case that the force F on a particle depends only on its position (coordinate). As an aside, when might the force depend on velocity? Conservative system?? If this is the case then one can prove the following: Let the forces acting on a particle be given by a vector F. dv If there exists a function V(x) such that F =, then the force field is said to be describable by the potential energy function, V(x). More generally, if there is a function V(x, y, z) such that V Fx,y,z ( ) = F i+ F j+ F k = x i V y j V x y z k = V(x, y, z) z then the force field F is said to be conservative. Here is the gradient operator and i, j and k are unit vectors in the x, y and z directions. Let's look at several examples to clarify this concept. 1. Gravitational potential for a particle of mass m: z m V(z) z 16

8 Since F = mgk, then V(z) = mg z. An ideal (Hookes Law) Spring Since F = kx i, we have V(x) = k x +C The quadratic potential energy curve also reminds us how we generally can visualize the forces acting on a particle. 17

9 How would we modify the potential energy for a diatomic molecule?? For the hydrogen atom? +e m r -e m e p The force is = qq 1 e F = attractive r r and the potential energy is V(r) = e, the Coulomb attractive Potential r 1

10 Schrödinger equation for a particle in one dimension Consider a particle of mass m confined to move in one dimension but with NO forces acting on the particle. Schrödinger postulated that the amplitude of the matter waves, ψ(x), must satisfy the differential equation, d ψ = Eψ m where E is the total energy and = h/ π More generally, if the particle is in a potential energy field V(x), then d ψ(x) + V(x) ψ(x) = E ψ (x) m Before developing formal postulates, let s see what this can mean. A solution to the first equation can be written in terms of complex exponentials or sines and cosines: ikx ψ(x) = e = cos kx + isin kx, where k = me This is a sine wave of wavelength λ = π/k. Since there are no forces (no potential energy), the total energy E is also the kinetic energy. Thus E = p /m and k= p/. The wavelength λ associated with the solution is thus h/p, the debroglie wavelength!! In the more general second case, the only difference is that E in the solution is replaced by E-V: m(e V) ikx ψ(x) = e = cos( kx ) + i sin( kx ), where k = The kinetic energy of the particle is (E-V), and the wavelength associated with the matter waves is now given by the relation λ = h m(e V) Thus, the slower the particle, the longer (and less wiggly) is the matter wave wavelength. Notice that when the particle is at rest, the wavelength is infinite, and we cannot tell where the particle is. This is the beginnings of the Uncertainty Principle at work! Now for the abstract Postulates for Quantum Mechanics. 19

df(x) = h(x) dx Chemistry 4531 Mathematical Preliminaries Spring 2009 I. A Primer on Differential Equations Order of differential equation

df(x) = h(x) dx Chemistry 4531 Mathematical Preliminaries Spring 2009 I. A Primer on Differential Equations Order of differential equation Chemistry 4531 Mathematical Preliminaries Spring 009 I. A Primer on Differential Equations Order of differential equation Linearity of differential equation Partial vs. Ordinary Differential Equations