Lecture 7 Schrödinger’s Equation and the Particle in a Box U= y(x) 0 L U= n=1 n=2 x n=3 Content Particle in a “Box” matter waves in an infinite square well Wavefunction normalization General p[.]
Lecture 7: Schrödinger’s Equation and the Particle in a Box y(x) U= n=1 n=3 U= L n=2 x Content Particle in a “Box” matter waves in an infinite square well Wavefunction normalization General properties of bound-state wavefunctions Last lecture: The time-independent SEQ (in 1D) KE term d y ( x) U ( x)y ( x) Ey ( x) 2m dx Total E term PE term Notice that if U(x) = constant, this equation has the simple form: d 2y C y( x ) dx where C 2m ( U E) is a constant that might be positive or negative For positive C, what is the form of the solution? a) sin kx b) cos kx c) eax d) e-ax For negative C, what is the form of the solution? a) sin kx b) cos kx c) eax d) e-ax Last lecture: The time-independent SEQ (in 1D) KE term d y ( x) U ( x)y ( x) Ey ( x) 2m dx Total E term PE term Notice that if U(x) = constant, this equation has the simple form: d 2y C y( x ) dx where C 2m ( U E) is a constant that might be positive or negative For positive C, what is the form of the solution? a) sin kx b) cos kx c) eax d) e-ax For negative C, what is the form of the solution? a) sin kx b) cos kx c) eax d) e-ax Constraints on the form of y(x) y(x)2 corresponds to a physically meaningful quantity – the probability of finding the particle near x Therefore, in a region of finite potential: y(x) must be finite, continuous and single-valued (because probability must be well defined everywhere) dy/dx must be finite, continuous and single valued (because dy/dx is related to the classical momentum) There is usually no significance to the sign of y(x) (it goes away when we take the absolute square) {In fact, we will see that y(x) can even be complex!} Exercise 1 Which of the following hypothetical wavefunctions for a particle in some realistic potential U(x) is acceptable? (a) y(x) (b) y(x) (c) y(x) x x x Which of the following wavefunctions corresponds to a particle more likely to be found on the left side? (c) (b) (a) y(x) y(x) y(x) x x x Solution Which of the following hypothetical wavefunctions for a particle in some realistic potential U(x) is acceptable? (a) (b) y(x) y(x) y(x) is not continuous at x=0 dy not defined dx y(x) x x (a) Not acceptable (c) (b) Acceptable Both y(x) and dy/dx are continuous everywhere x (c) Not acceptable dy/dx is not continuous at x=0 Solution Which of the following wavefunctions corresponds to a particle more likely to be found on the left side? (c) (b) (a) y(x) y(x) y(x) x 0 x x None of them! (a) is clearly completely symmetric (b) might seem to be “higher” on the left than on the right, but it is only the absolute square the determines the probability y2 x Application of SEQ: “Particle in a Box” Recall, from last lecture, the time-independent SEQ in one dimension: d y ( x) U ( x)y ( x) Ey ( x) 2m dx KE term Total E term PE term As a specific important example, consider a quantum particle confined to a small region, < x < L, by infinite potential walls We call this a “one-dimensional (1D) box” U(x) This is a basic problem in “Nano-science” It’s a simplified (1D) model for an electron confined in a quantum structure (e.g., “quantum dot”), which scientists/engineers make, e.g., at the UIUC Microelectronics Laboratory! (www.micro.uiuc.edu) ‘Quantum dots’ L U = for < x < L U = everywhere else (www.kfa-juelich.de/isi/) (newt.phys.unsw.edu.au) Waves: Boundary conditions Boundary condition: Constraints on a wave where the potential changes E = at surface of a metal film Displacement = for wave on string E=0 If both ends are constrained (e.g., for a cavity of length L), then only certain wavelengths l are possible: n L l 2L L 2L/3 L/2 nl = 2L n = 1, 2, … ‘mode index’ Rope Demo ...Content Particle in a ? ?Box? ?? matter waves in an infinite square well Wavefunction normalization General properties of bound-state wavefunctions Last lecture: The time-independent SEQ (in 1D)... eax d) e-ax Constraints on the form of y(x) y(x)2 corresponds to a physically meaningful quantity – the probability of finding the particle near x Therefore, in a region of finite potential:... “higher” on the left than on the right, but it is only the absolute square the determines the probability y2 x Application of SEQ: ? ?Particle in a Box? ?? Recall, from last lecture, the time-independent