8.
04 Quantum Physics
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Since V (x) = 0 in the region where the particle can be found, the zero-point energy must be purely kinetic in this case. We could have estimated it using the Heisenberg uncertainty relation: Conning the particle to a region x = a introduces p)2 h h2 a momentum uncertainty p = h , and a kinetic energy ( 2ma 2 . Since x a 2m p2 h2 2 V (x) = 0 for x x a, and 2m x2 = E in this region, and E = 2m in CM, it appears that momentum is associated with the derivative x of the wavefunction, 2 and kinetic energy with the curvature x2 . The eigenfunctions have the property that a x 2 x dx sin n sin m (11-1) dxun (x)um (x) = a a a 0 1 a x x = dx cos (n m) cos (n + m) (11-2) a 0 a a sin((n m) ) sin((n + m) ) = (11-3) (n m) (n + m) m, 0 for n = = (11-4) 1 for n = m. = nm nm = m, 0 for n = Kronecker delta 1 for n = m. (11-5)
(11-6)
This is a general property, not particular to this example: Eigenfunctions belonging to dierent eigenvalues are orthogonal, if the eigenfunctions are normalized we call them orthonormal. dxu orthonormality condition (11-7) n (x)um (x) = mn
Complex conjugate not necessary for box potential, where eigenfunctions are real, but necessary in general. un un dx = |un |2 dx = 1 normalization (11-8)
Eigenfunctions as basis
Why is knowing the eigenfunctions important? Consider box potential: Fouriers theorem states that any function (x) that that boundary conditions (0) = satises x 0 = (a) can be written as a sum of sin n a . Massachusetts Institute of Technology XI-1
8.04 Quantum Physics
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Note. cos n x terms do not appear since they do not satisfy the boundary condia tions. Since un sin n x , we can also write a (x) =
n=0
cn un (x)
expansion of arbitrary (x) into eigenfunctions
(11-9)
In order to calculate the expansion coecients cn , we use the orthonormality property of the eigenfunctions. a dxu dxum (x) cn un (x) (11-10) m (x) =
0 n=0
n=0
cn
0
dxu m un (x)
n=0
cn mn
(11-11) (11-12)
= cm Thus the expansion coecient can be calculated as the integral cm =
dxu m (x) (x)
expansion coecients
(11-13)
Again we do not need the complex conjugate here, since um is real, but we have written it in the correct general form. We call the set of functions {un (x)} complete if an arbitrary function (x) can be written as a superposition of functions of the set. A complete, orthonormal set of functions is called a basis. The above properties, derived for the particles in the box, are true in general in QM: 1. The energy eigenfunctions un of a Hamiltonian H= h 2 2 + V (x) 2m x2 (11-14)
form a basis, an arbitrary wavefunction (x) can be expanded as superposition of eigenstates (x) = cn un (x) (11-15)
n
with complex coecients cn . If the spectrum of eigenvalues, or part of the spectrum of eigenvalues is continuous, the expansion contains an integral (x) = dEc(E )uE (x). n cn un (x) + Massachusetts Institute of Technology XI-2
8.04 Quantum Physics 2. The expansion coecients cn are given by cn =
Lecture XI
dxu n (x) (x) ,
(11-16)
i.e., they can be calculated once the eigenfunctions un are known explicitly.
Analogy to vector analysis
Consider an n-dimensional vector space. A set of n mutually orthogonal unit vectors {e i }i=1,...,n forms a basis, i.e. an arbitrary vector v can be expanded into unit vectors: v=
n
i=1 complex basis number vector
ci e i
(11-17)
with suitably chosen coecients ci . The ci are uniquely determined, and given by ci = e i v. (11-18)
In vector analysis terms, the wavefunctions (x) form a vector space, called the
Figure I: projection of v onto e i Hilbert space, the energy eigenfunctions un (x) form a basis. The dimension of the Hilbert space is the number of independent energy eigenfunctions; if that number is innite, the Hilbert space is innite-dimensional. We have the following correspondences: Massachusetts Institute of Technology XI-3
8.04 Quantum Physics QM wavefunction: (x) energy eigenfunction: ui (x) # of independent ui (x) dxu i (x) (x) dx1 (x)2 (x) dxu i (x)uj (x) = ij (x) = i ci ui (x) for any (x) Eigenvalue equation: E (x) = EE (x) H Set of eigenfunction of certain type of operators (Hermitian operators) forms basis. Hamiltonian is such an operator. vector: v basis vector: e i dimension of vector space dot product: e i v dot product: v1 v2 vector analysis
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orthonormality: e i e j = ij Completeness: v = i ci e i for any v Eigenvalue equation: v = mv M Set of eigenfunctions of certain types of matrices forms basis. (self-adjoint matrices M = M .)
Physical interpretation of expansion coecients
Assume that we have prepared some arbitrary wavefunction (x) (that is consistent with the coundary conditions) inside the box.
Figure II: Pictorial expansion of wavefunction in a box in terms of eigenfunctions.
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8.04 Quantum Physics Expansion into eigenfunctions: (x) =
i=1
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ci ui (x)
(11-19) (11-20)
= c1 u1 (x) + c2 u2 (x) + . . .
phase (sign) of ci important, detemines whether more amplitude on left/right etc. The larger |ci |, the more the wavefunction (x) is like ui ( x), (the larger the projection of (x) onto ui , given by (analogy to e i v) ci = dxu i (x) (x)). 2 Calculate i=1 |ci | :
i=1
|ci |2 = = =
i=1 i=1
c i ci c i dxu i (x) (x)
c i ui (x)
(11-21) (11-22) (11-23) ci ui (x) (11-24) (11-25) (11-26) (11-27)
dx (x) = = = =1 dx (x)
i=1 i=1
dx (x) (x) dx| (x)|2
2 2 Since i=1 |ci | = 1, the quantity |ci | can be interpreted as the probability to nd the particle in the state i, if a measurement of the particles energy eigenstate is made, given that the particle has been initially prepared in a state characterized by an arbitrary wavefunction (x). How is a measurement of the energy eigenstate of the particle made? = Measure energy of particle. h2 2 En = n2 , = E1 = 2 ma2
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Figure III: Measurement of particle energy in state that is not an energy eigenstates can yield dierent values Ei with probabilities |ci |2 .
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