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Unformatted text preview: Position Representation, Time Independent Schrodinger Equation, Particle-in-box
Chem 120A 02/14/07 Spring 2007
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; Lecture 13 1 Position Representation of Quantum State Function
We will motivate this using the framework of measurements. Consider ﬁrst the simpler example of a photon. The polarization of the photon can be either horizontal (H ) or vertical (V ), from which we have a discrete basis of two states H and V . We can measure the polarization by passing the photon through a polarizer crystal, which passes either H or V light depending on its orientation. The measurement operators for this simple 2-state basis are MH = H H , MV = V V. A single measurement on an arbitrary state ψ will collapse ψ onto one of the two orthonormal basis vectors. For example, if the H measurement is made, the state after measurement will be Hψ . H ψH Hψ If the measurement is repeated many times, this state will be obtained with probability PH = | H ψ |2 . Now consider a particle in a quantum state, e.g., the energy level of a hydrogen atom. The hydrogen atom consists of 1 positively charged proton in the nucleus and 1 negatively charged electron. The electron is ∼ 1800 times lighter than the proton, so to a ﬁrst approximation the electron can be regarded as moving around a stationnary proton. The possible energy levels for this electronic motion form a discrete, inﬁnite set of levels of negative total energy (indicating overall binding to the proton), and are given by the relation En ∼ −1/n2 , n = 1, 2, 3, ...... The energy eigenvectors n formed by these energy levels form an inﬁnite dimensional Hilbert space. Now what if we want to observe the electron? It is moving in conﬁguration space, so lets consider the effect of the measurement operator corresponding to a location r in conﬁguration space. The measurement operator is Pr = r r r ψ , with probability and a measurement on the ket ψ collapses this onto the state r | r ψ |2 = |ψ (r)|2 .
Chem 120A, Spring 2007, Lecture 13 1 So |ψ (r)| is the probability amplitude of ﬁnding an electron at r, i.e., “the wave function in the position representation”. Note that the state after measurement is the position ket r . We can understand this in a pictorial manner by imagining a basis consisting of a very densely spread set of delta functions: the wave function is the amplitude of the the expansion of the quantum state in this basis. ψ
rψ = = ∑ αi ri ∑ αi
i i r ri = αi δ (r − ri ) = ψ (r). The position representation is deﬁned by the continuous set of basis vectors r , satisfying dr r r = 1 (completeness) = δ (r − r′ ), r r′ where δ (r − r′ ) is the Dirac delta function. This is deﬁned by the relation (shown here for 1D)
+∞ −∞ dxδ (x − x′ ) f (x′ )dx′ = f (x). Setting f (x) = 1 shows that the integral under the delta function is equal to unity. The three dimensional delta function is given by δ (r − r′ ) = δ (x − x′ )δ (y − y′ )δ (z − z′ ).
We can regard the Dirac delta function as the limit of a sequence of functions possessing unit norm, e.g., a sequence of Gausssians with variable width λ : fλ =
′2 1 2 √ exp−(x−x ) /2λ . λ 2π Note that the norm of the basis states r is ill-deﬁned, unless one agrees to implicitly integrate over the position coordinate and make use of the delta function property. To summarize, the ket ψ can be expanded in the position representation as ψ= d r′ r′ r′ ψ and φ can be expressed in terms of the corresponding wave The inner product between two state ψ functions in the position representation: φψ = = dr φ r r ψ d rφ ∗ (r)ψ (r).
2 Chem 120A, Spring 2007, Lecture 13 Now the norm is well-behaved ψψ = ψ ∗ (r)ψ (r)d r = 1. This implies we can choose a set of functions φn (r) satisfying
∗ φn (r)φm (r)d r = δmn which is just the orthonormality condition between φn and φm . We can make this set of functions a basis for the Hilbert space spanned by the energy eigenstates n . This basis of wave functions in position representation has a well behaved norm ||φn ||2 = |φn (r)|2 d r = 1. These functions are therefore a set of square integrable functions, often also called L2 functions. Similar arguments lead to the deﬁnition of the momentum representation. The ket ψ can be expanded in the momentum representation as ψ= d p′ p′ p′ ψ where p′ ψ = ψ (p′ ) is the probability amplitude to ﬁnd the particle with momentum p′ . It is the wave function in the momentum representation. Note that equivalently, it can be understood as the expansion coefﬁcient in the expansion in momentum eigenstates p′ . Projecting this expansion into the position representation yields the basic equation relating position and momentum representations of a quantum state ψ : ψ (r) = r ψ = dp′ r p′ ψ (p′ ). Note that using the Dirac notation we are correct in writing ψ on both right and left hand sides of this equation. However, the two functions may have very different dependence on their respective variables r ˜ and p. To avoid confusion, one usually gives these different names, e.g., ψ (r) and ψ (p). Transformation between position and momentum representations What is the transformation element r p′ in the above equation? If we set this equal to eip·r then the ˜ equation looks like a Fourier transform of the wave function in momentum space, ψ (p), i.e., ψ (r) = ˜ d peip·r ψ (p). This is not quite a Fourier transform, since we have momentum p rather than wave vector k in the integral. However, p and k satisfy the de Broglie relation, p = hk ¯ which leads to the Fourier transform relation ψ (r) = ˜ d keik·r ψ (k) where we have omitted factors of h and 2π . ¯
Chem 120A, Spring 2007, Lecture 13 3 2 Solving the Time independent Schrodinger Equation
First, in general, as a matrix equation. Remember that the Hamiltonian gives energy as the eignenvalue when it acts on a eigenstate. H Ψ =E Ψ We will assume we know an orthonormal basis n with n = 1...N and m n = δmn . We can then expand our state in that basis. Ψ = ∑n n H ∑n n nΨ nΨ n Ψ = E ∑n n Taking the inner product m H ∑n n mHn n Ψ = E ∑n m n n Ψ n Ψ =E m n n Ψ This is an eigenvalue Matrix equation 1H1 . . . . . NH1 ..... 1HN . . . . . NHN 1Ψ . . . . . NΨ 1Ψ . . . . . NΨ ..... =E We ﬁrst diagonalize the basis matrix and ﬁnd the energy eigenvalues. We can then plug the eigenvalues in to get the eigenvectors. So in general if we can pick the right basis, we can solve for the eigenvalues (energies) and eigenvectors (states) using the Matrix equation above. We will deal with such problems later. However, ﬁrst, we will look at a simpler system where our basis is composed a position eigenspace in one dimension (position representation). Position is continuous, so instead of summing over all basis states in our expansion, we will take the integral.
∞ −∞ ˆ ˆ ¯ ˆ¯ x ( 2p + V (x) x d x x Ψ = x Ψ E m x Ψ ≡ Ψ(x) Leading to the 1-D, time-independent Schrodinger Equation Chem 120A, Spring 2007, Lecture 13 4 −h2 d 2 ¯ 2m dx2 Ψ(x) + V (x)Ψ(x) = E Ψ(x) Rearranging this equation
d2 Ψ(x) dx2 = 2m (V (x) − E )Ψ(x) h2 ¯ Similar to the Matrix equation above, we would like to solve for both the energy as well as the wavefunction. These will depend on the given potential which conﬁne our quantum particles. 3 Particle-in-a-box
We will now solve the time-independent Schrodinger Equation for a simple problem relevant to many chemical and physical systems, a particle conﬁned to a ﬁnite region (box). The model is described in Figure 1 below. The potential is 0 when the particle is between 0 and a. The potential is V0 = ∞ when the particle is not in this region. Since the energy becomes inﬁnite outside the region, there can be no solution having non-zero probability amplitude Ψ(x) outside the box. We say that the particle is constrained to the above potential (V(x) = 0 for 0 < x < a. V(x) = infinity for x < 0 V(x) = infinity for x > L particle never here wavefunction = 0 for x < 0 particle lives in here wavefunction = 0 particle never here wavefunction = 0 for x < 0 V(x) = 0 x=0 x x=L Figure 1: Particle in a box This is a reasonable model for an electron in the lowest few levels of the hydrogen atom, and for π -electrons in conjugated molecules. Application to the former is discussed in the additional notes ”Quantum Mechanics in a Nutshell II”. Application to the latter will be described at the end of the notes for the next lecture (14). Chem 120A, Spring 2007, Lecture 13 5 ...
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