2. Time-Independent Schrödinger Equation Background A course covering modern physics A course in intermediate mechanics Harris, Non-Classical Physics Paul A. Tipler, Ralph A. Llewellyn, Modern Physics Hermite polynomial and function handout Definite integrals handout Fourier transform section of the integral transform handout Concepts of primary interest: Schrödinger equation Separation Ladder operations – raising and lowering Expectation values Phase and group velocities Motivation of the momentum operator Sample calculations: SC1 - Griffiths Example 2.2; eigenstate expansion [SE(ti).7] SC2: Square well mixed state probability density [SE(ti).8] SC3: Raising o to find 1 [SE(ti).22] SC4: Expectation values using ladder operators [SE(ti).25] SC5: Bound state for a delta potential: - (x – xo) [SE(ti).26] SC6.) Free particle scattering by a delta potential [SE(ti).30] SC7.) The step potential [SE(ti).33] SC8.) The high step potential [SE(ti).36] SC9.) The finite square well [SE(ti).37] SC10.) Barrier penetration [SE(ti).41] Helpful handouts: Hermite Polynomials Tools of the trade: This handout is keyed to Griffiths Introduction to Quantum Mechanics, 2nd Ed. It is not designed to be used independently. The wave function: A quantum system is described by a wave function which contains all the information that can be known about the system. It is a function of the generalized coordinates for the system and time. In 1D, the function might have the form (x,t). It is a complex-valued function. for use with Griffiths QM Contact: [email protected] Schrödinger proposed the equation that governs the time-development of the wave function.1 22 ˆ i (,) xt H (,) xt2 V () x (,) xt [SE(ti).1] tmx2 22 The operator 2 Vx() is an example hamiltonian for a one dimensional 2mx system. At least in simple cases, the hamiltonian represents the energy of the system. Expressing [SE(ti).1] as an eigenvalue equation yields: 22 2 Vx() nnnn (,) xt E (,) xt i (,) xt [SE(ti).2] 2mx t To invoking the separation hypothesis, assume: n(x,t) = n(x) Tn(t). 1122 2 Vx() nn () x E i Tt() n ()xmx 2 Ttt() 22 Time independent SE: 2 Vx()nnn (,) xt E () x 2mx En it Time dependence: E Tt() i Tt () Tt() e nn t n n En it Energy eigenfunctions: n(x, t) = n(x) e Each energy eigenvalue solution has a stationary probability distribution. In summary, the hamiltonian is the time development operator, and it is the energy operator for essentially for all systems that we shall study. After separation, the focus of each new problem exploration is to find the physically acceptable solutions of the time-independent Schrödinger equation. 1 See the article by Felix Bloch in the December 1976 issue of Physics Today. 9/17/2010 SP425 Notes –Time Independent Schrödinger Equation Ch2-2 22 2 Vx()nn (,) xt En () x [SE(ti).3] 2mx The set of values of the constant En for which the equation has an acceptable solution are the energy eigenvalues for the problem, and the corresponding states described by the n(x) are the eigenstates. When combined with the time dependences, the full En i t wavefunctions n(x, t) = n(x) e are a complete (basis) set for the expansion of any permissible solution to the problem. That is: for a general (x, t), (x ,0)axkk ( ,0) akk (x ) kk11 itk (,)xt akk (,) xt akk () xe [SE(ti).4] kk11 The solutions corresponding to distinct eigenvalues are necessarily orthogonal, and those corresponding to degenerate eigenvalues can be chosen to be orthogonal (or made orthogonal using a method like Gram-Schmidt). *()xxdx () [SE(ti).5] km k m km Note that [SE(ti).5] also states that (x) is normalized. Beware: we can only normalize bound-state wavefunctions. A plane wave like A eikx does not represent a bound state, and it cannot be normalized(using our bound-state convention). You will generalize the definition of normalized when you reach graduate school. The problem disappears. Given an orthogonal basis, the expansion coefficients can be found using projection. To find ap, take the inner product of the corresponding behavior p and [SE(ti).4] using the orthogonality [SE(ti).5]. p ()xx (,0) pk()x (,0)xax p ()k ()x ap [SE(ti).6] k1 pp()x ()x 9/17/2010 SP425 Notes –Time Independent Schrödinger Equation Ch2-3 In the case of quantum mechanics, the abstract inner product represents a concrete integration. ()x (,0)xx [ ()](,0)* xdx. pp upper BRA KET [] BRA* KET dx lower Erwin Rudolf Josef Alexander Schrödinger (August 12, 1887 – January 4, 1961) was an Austrian - Irish physicist who achieved fame for his contributions to quantum mechanics, especially the Schrödinger equation, for which he received the Nobel Prize in 1933. In 1935, after extensive correspondence with personal friend Albert Einstein, he proposed the Schrödinger's cat thought experiment. The huge Schrödinger crater on the far side of the Moon was posthumously named after him by the IAU. The Erwin Schrödinger International Institute for Mathematical Physics was established in Vienna in 1993. http://en.wikipedia.org/wiki/Schroedinger The infinite square well: A few observations about the character of the solution of the infinite square well that are common to quantum problems are to be noted as well as one our two that are peculiar to problems like the infinite square well. forbidden 0inside Vx() outside forbidden The energy must be greater than zero (or, in general, E > Vmin) in order that an acceptable (normalizable) solution exist. 9/17/2010 SP425 Notes –Time Independent Schrödinger Equation Ch2-4 The spatial part of an energy eigenvalue solution can be expressed as a real-valued function of an oscillatory form in the classically allowed regions. (Oscillatory form means that the function is concave toward zero.) Interlaced Zeros Property: In one dimension, if a reference spatial eigenfunction has n internal* zero crossings, the state with the next higher eigen-energy has n + 1 zero crossings with one crossing located between each adjacent pair of zero crossings of the state with n crossing. * Internal: those not at an endpoint. Here, crossings for xmin < x < xmax. Endpoint zeros may be common to all allowed states for a problem. The infinite square well potential is symmetric about the center of the well. The eigenstates are alternately even and odd about the center of the well. The eigenstates are normalizable, and they are non-zero only inside the well. As they are continuous functions, the eigenstates have a zero at each edge of the well. The eigenstates form a complete, orthogonal basis for the expansion of normalizable spatial functions in the region inside the well (in the region in which they are non-zero). The expansion coefficients can be found using projection. For a general (x,0), you take the inner product with k| to find ck. (,0)x cxmm () k()xx (,0) cmkm () x () xwherex km() () x km * ()xx (,0) k ()xxdx (,0) cxk inside *() (,0)xdx k * inside k km()xx () ()xxdx () inside kk We assume that the functions k(x) are normalized. 9/17/2010 SP425 Notes –Time Independent Schrödinger Equation Ch2-5 If, at time t = 0, the wave function of the system is (x, 0), then Em (,)x tcxe ()it() where c* () xx (,0)dx mm kinside k Exercise: Consider each fact presented bout the infinite symmetric square well. Discuss the validity of each for a finite symmetric square well problem. Properties of acceptable wave functions: The presence of the second derivative with respect to position in the energy operator (the hamiltonian) requires that (x) is continuous, and it normally requires that x is continuous. The wavefunctions for the infinite well problem are continuous as is always required. A discontinuous first derivative x leads to a singular (ill-defined, infinite) second derivative. The infinite well hamiltonian has a singular potential, and its wavefunction can have a discontinuous first derivative at points for which the potential is singular. Not that the potential has an infinite discontinuities at the points for which x is discontinuous. n 2m Vx() E () x xx 2 nn For a particular infinite depth square well with the zero-potential region running from 0 to a, the eigenfunctions are: En 22 (xt , )2 sin[nx ]ei( )t where E n2 nnaa 2ma2 The particles are confined in a region of width a, and the spacing between the energy -2 levels is proportional to a . A narrower confinement region leads to greater spacing between allowed energies. In the limit of large a (no confinement), there would be a continuum of allowed energies rather than a spectrum of discrete values. The n2 9/17/2010 SP425 Notes –Time Independent Schrödinger Equation Ch2-6 dependence of the energy on the quantum number is characteristic of a particle confined in a region of fixed width. Sample Calculation SC1 - Griffiths Example 2.2: Infinite Square Well Problem Consider a square well with the zero-potential region running from 0 to a. At time 0, the spatial wave function is: (x, t) = A x (a – x). [SE(ti).7] Note that this wave function respects the conditions required for an admissible wavefunction. It meets the boundary conditions as it vanishes at 0 and at a. It is normalizable. (Evil things can happen when the wavefunction has no respect!) You should always verify that a wavefunction to be studied is normalizable and that it meets the boundary conditions for the problem. If it does not, you may proceed, but you should be prepared for nonsense to follow. a.) Normalize the function. b.) Expand the function in terms of the eigenfunctions.