The Schrodinger Wave Equation (Engel 2.4) In QM, the behavior of a particle is described by its wave function Ψ(x,t) which we get by solving:

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1 When do we use Quantum Mechanics? (Engel 2.1) Basically, when λ is close in magnitude to the dimensions of the problem, and to the degree that the system has a discrete energy spectrum The Schrodinger Wave Equation (Engel 2.4) In QM, the behavior of a particle is described by its wave function Ψ(x,t) which we get by solving: Time-dependent SWE = x J-s =Hamiltonian operator=h When V(x,t) =V(x); i.e., not a function of time Time-independent SWE

2 What does have the solution Ψ(x) do for me? (Engel 3.1) Born statistical interpretation of the wave function says that Ψ(x,t) 2 gives the probability of finding the particle at point x, at time, t. = probability of finding the particle between x and x+dx at time t. Low probability High probability

3 The Born interpretation introduces indeterminacy into QM; even if you know everything about the particle (it s wavefunction) QM can only offer statistical information about the possible results. Max Born We will return to this but for now let s talk more about the Schrodinger equation and the basics of QM Formalism of Quantum Mechanics Table with function, bra-ket and vector notation

4 QM Bra-ket Vectors f(x) f v A f x =g x A f>= g> A v=w A af+bg A au+bv =aaf+bag f x g x dx f x A g x dx g x A f x dx = Af x g x dx Hermitian <f g> f Ag g A f Complex conjugate of 1 st term is implied =aa u+ba v u v u A v = Af g

5 A Hermitian matrix is one with elements that are symmetric about the diagonal and are each other complex conjugate Formalism of Quantum Mechanics Operators An operator is a transformation that takes a function as an input and produces another function (usually). Example: In QM, most operators are linear: a, b constants Multiplication by x: The hat ^ is usually there to show that it is an operator. For example the total energy is called a Hamiltonian, H, is one formulation of CM by Hamilton. In QM H the Hamiltonian for the total energy is an operator. Very important (later)

6 See Engel Table 3.1 for some of the most important QM operators (observables): p. 41 Order matters! Also Example: In QM an observable; i.e., any quantity that can be measured has an associated linear operator. Any values that are constant for every measurement correspond to the eigenvalues of the operator for that quantity.

7 Eigenvalues and eigenfunctions In general, an operator operating on a function gives a different function In certain cases, the operator operating on a function gives the same function multiplied by a constant. That function is an eigenfunction of the operator, and the constant is an eigenvalue. These are important in measurements. = an eigenvalue equation Example: What about e ax2? No!

8 Example: The eigenvalue (-a 2 ) has more than one eigenfunction. This is related to the concept of degeneracy (later). To solve an eigenvalue equation multiply by f * and integrate: if f is normalized (later)

9 Example: Show that is an eigenfunction of and find its eigenvalue The function is an eigenfunction if it obeys: Eigenvalue = -1

10 Hermitian Operators Hermitian operators are those operators with real eigenvalues. In QM they represent observables because the outcome of a measurement is real Definition: An operator is Hermitian if Example: is d/dx Hermitian for eigenfunction: e ikx? No because the eigenvalue is imaginary.

11 Show that all the eigenvalues of a Hermitian operator are real. If A is Hermitian Only possible if a = a *

12 Example: Consider the fact that all wavefunctions in QM are square integrable meaning they go to 0 at the limits of their defined intervals Is a Hermitian operator for functions on the x-axis which go to 0 at infinity? Write down the definition of Hermitian for the operator: LHS Integration by parts =RHS

13 It can deduced therefore that p x is Hermitian for the space of all functions that are square integrable. Orthogonality (Engel 2.6) In a 3D vector space 2 vectors are orthogonal if the angle between them is 90 o (also called perpendicular) Their dot product Two functions are orthogonal if Note: Dirac bra-ket notation. For a Hermitian operators there eigenfunctions ø i (x), i = 1, 2, such that

14 Example: Show that the set of functions: is orthogonal if m and n are integers Show that a) If m = n b) If m n So they are orthogonal. See problem 2.6 in Engel.

15 Show that the eigenfunctions corresponding to a Hermitian operator are orthogonal Let Ω be an operator with eigenfunctions f, g with eigenvalues ω 1 and ω 2 But Hermitian means that: Real eigenvalues for Hermitian operators

16 But f(x) and g(x) are orthogonal Normalization A function is said to be normalized if To normalize a function is to scale it so that it is normalized. This comes from the Born statistical interpretation of the wavefunction. If a set of functions is orthogonal and each function is normalized the set is said to be orthonormal Kronecker delta function

17 Example: Normalize the function on the interval (0, ) Example: Normalize the function over the interval 0 x a

18 The need to normalize wave functions will become apparent when we discuss the meaning of wave functions in QM: The square of the wave function is a measure of the probability of finding the particle in a certain location. Since the particle has to be somewhere the integral of the probability over the entire space must be 1 (i.e. 100%).