* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download No Slide Title
Bell's theorem wikipedia , lookup
Bohr–Einstein debates wikipedia , lookup
Wheeler's delayed choice experiment wikipedia , lookup
Quantum decoherence wikipedia , lookup
Renormalization group wikipedia , lookup
Coherent states wikipedia , lookup
Hydrogen atom wikipedia , lookup
Symmetry in quantum mechanics wikipedia , lookup
Quantum state wikipedia , lookup
Dirac equation wikipedia , lookup
Double-slit experiment wikipedia , lookup
EPR paradox wikipedia , lookup
Wave–particle duality wikipedia , lookup
Canonical quantization wikipedia , lookup
De Broglie–Bohm theory wikipedia , lookup
Density matrix wikipedia , lookup
Erwin Schrödinger wikipedia , lookup
Identical particles wikipedia , lookup
Molecular Hamiltonian wikipedia , lookup
Schrödinger equation wikipedia , lookup
Wave function wikipedia , lookup
Measurement in quantum mechanics wikipedia , lookup
Matter wave wikipedia , lookup
Ensemble interpretation wikipedia , lookup
Hidden variable theory wikipedia , lookup
Quantum electrodynamics wikipedia , lookup
Path integral formulation wikipedia , lookup
Many-worlds interpretation wikipedia , lookup
Particle in a box wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
Interpretations of quantum mechanics wikipedia , lookup
Lecture 2: The Born Interpretation of the Wavefunction This lecture covers the following parts of Atkins 11.4 The Born interpretation of the wavefunction Lectures on-line Born Interpretation (PDF) Born Interpretation (PowerPoint) Tutorials on-line The postulates of quantum mechanics (This is the writeup for Dry-lab-II)( This lecture coveres parts of postulate 2) Probability The wave function and probability The Development of Classical Mechanics Experimental Background for Quantum mecahnics Early Development of Quantum mechanics Audiovisuals on-line review of the Schrödinger equation and the Born postulate (PDF) review of the Schrödinger equation and the Born postulate (HTML) review of Schrödinger equation and Born postulate (PowerPoint **, 1MB) Slides from the text book (From the CD included in Atkins ,**) Quantum Mechanical Hamiltonian From Last Lecture ˆ is constructed by the The quantum mechanical Hamiltonian H following transformations: 1 ˆ HClass H pˆ x2 pˆ 2y pˆ z2 V( xˆ , yˆ , zˆ ) 2m x xˆ ; y yˆ ; z zˆ ; i x py pˆ y i y px pˆ x pz pˆ z i z 2 2 2 ˆ H [ 2 2 2 ] V(x, y,z) 2m x y z 2 Solve Schrödinger equation ˆ = E H to obtain Quantum Mechanical Hamiltonian Postulate 2 To every physical observable class there corresponds a linear QM Hermitian operator. In order class (x,y,z, px ,py,pz ) to obtain this operator, we write out the ˆ QM ( xˆ , yˆ , zˆ , pˆ x , pˆ y,pˆ z ) classical - mechanical expression for the observable in terms of Cartesian coordinates and corresponding linear momentum components and perform the following transformations : xˆ x ; yˆ y ; zˆ z ; i x py pˆ y i y px pˆ x pz pˆ z i z Interpretation of the wavefunction in 1- D ˆ = E Schrödinger Equation H 1. The wavefunction contains all the dynamic information about the system it describes 2. The square modulus of the wavefunction at x is proporional to the probability of finding the particle at x If the wavefunction of a particle has the value (x) at a point x,than the probability of finding the particle between x and x + x is : P (x) (x)dx * Max Born Interpretation of the wavefunction in 1- D (x) probability amplitude positive, negative, complex | (x) |2 (x)(x)* always positive c = a + ib a, b real cc (a + ib)(a + ib) (a + ib)(a - ib) * P (x) (x)dx * = a 2 - iab + iab + b 2 * Interpretation of the wavefunction in 1- D (x) probability amplitude positive, negative, complex 2 | (x) | (x)(x)* always positive P (x) (x)dx * Interpretation of the wavefunction in 3- D If the wavefunction of a particle has the value (x, y, z) at a point (x,y, z) than the probability of finding the particle between x and x + x; y and y + y; z and z + z is : dz dy dx P(x, y, z) (x, y, z) (x, y, z)dxdydz * Probability Probability density= Probability per volume Volume element unit QuickTime™ and a Video decompressor are needed to see this picture. Spherical Coordinates We are going to make use of the spherical polar coordinate system Longitude Latitude Distance Spherical Coordinates We are going to make use of the spherical polar coordinate system Z (x,y,z) (r, r X Y We have the following relation x= r sincos y= r sinsin z= r cos Spherical Coordinates The volume between (r, , ) and (r + r, + , ) r r rsin rsin r r sinr 2 Volume element Spherical Coordinates Z (x,y,z) (r, r X Y The Laplacian is in C artesian coordinates given by 2 d2 d2 d 2 = [ 2 + 2+ 2 ] dx dy dz In po lar sphe rical coordinates we ge t "after some rather comp licated calculation s " d2 2 d 1 d2 1 d 1 d2 2 =[ 2 + r + 2 + 2 cot + 2 2 ] 2 d r d d dr r d r r sin Normalization If yis a solution to the time-independent Schrödinger equation 2 y(x) y (x)V(x) Ey(x) (1) 2 2m x 2 then Ny will also be a solution Proof: 2 2 (Ny(x)) Ny(x)V(x) ENy (x) (2) 2 2m x or (Ny(x)) [ ] Ny(x)V(x) ENy(x) 2m x x 2 Normalization (Ny(x)) [ ] Ny(x)V(x) ENy(x) 2m x x 2 y(x) [N ] Ny(x)V(x) ENy(x) 2m x x 2 2 y (x) N Ny(x)V(x) ENy(x) 2 2m x 2 Thus Eq.(2) is equivalent to Eq.(1) Normalization However we shall shortly show that of all the solutions Ny the only acceptable is the one that satisfy : x b y( x)y( x)* Ny (Ny ) dx 1 * x a a b For a wavefunction defined in Normalized interval a to b Thus : N 1 x b 1/2 y ( y ) dx 1 x a * Normalization The normalization of N is required to satisfy the Born interpretation of the function N y ( x) Born interpretation: x xx P(x) Ny(Ny ) x P( x ) x x * P(x) Ny(Ny ) x * a x x+x * N y ( N y ) dx b Probability of finding particle between x and x Normalization and Born Interpretation By dividing the interval between a and b into sections y( x)y( x) * P(xi ) N 2 (xi )(xi )* dxi P(x1 ) P(xi ) P(xk ) xi We have that the probability of finding the particle in its Possible range between a and b is: P(xn ) x1 a a xi xk xn b b i n Pab P(xi ) i 1 Normalization and Born Interpretation y( x)y( x) * i n Pab P(xi ) i 1 Pab N (xi )(xi ) xi 2 xi x1 a a in * i1 x i xk x n Pab N (xi )(xi ) dx i 2 b b b a * Normalization and Born Interpretation y( x)y( x) * The probability of finding the Particle between a and b b Pab N (xi )(xi ) dx i 2 * a xi Is 100 % x1 a x i xk x n Thus by convention b b N (xi )(x i ) dxi 1 2 a b * a Probabilities are out of a total of 1 Generalization to 3D QuickTime™ and a Video decompressor are needed to see this picture. Properties of and Born Interpretation The Born intepretation imposes a number of restrictions on Not continuous 2 so and 2 not defined x x Schrödinger equation H = E not defined 2 (x) V(x)(x) E(x) 2 2m x 2 Normalization and Born Interpretation Not continuous x so 2 x 2 not defined Schrödinger equation H = E not defined 2 (x) V(x)(x) E(x) 2 2m x 2 Normalization and Born Interpretation P(x) Ny(Ny ) x * not single valued Not permissable b N (xi )(x i ) dxi 2 * a Not permissable Postulate 1. QM Postulate 1. The state of a quantum system is described by a function (x, y, z, t) or (r, t), of the configuration space variables and time. The function is referred to as the state function or the wavefunction. It contains all the information that can be determined about the system. Furthermore , we require that (x, y, z, t) be single valued, continuous, differentiable to all orders, and quadratically integrable Comments to Postulate 1 QM Postulate 1. The wavefunction itself has no physical interpretation since it can be a complex function However, the function * 2 (r,t)(r,t) | (r,t) | , does have an interpretation as a probability density and, in principle, is a measurable quantity What you should know from this lecture Born postulate in 1D If the wavefunction of a particle has the value (x) at a point x,than the probability of finding the particle between x and x + x is : P (x) (x)dx * Born postulate in 3D If the wavefunction of a particle has the value (x, y, z) at a point (x,y, z) than the probability of finding the particle between x and x + x; y and y + y; z and z + z is : P(x, y, z) (x, y, z) (x, y, z)dxdydz * What you should know from this lecture You are not required to derive or remember the expression for the Laplacian or the volume element in spherical coordinates. However you should know the definition of the three variables r,, and their relations to x,y, z You should know how to normalize a function You should understand why the integral of the square modules of a wavefunction integrated over all space in which the wavefunction is defined must give 1 according to Borns postulate You should be familiar with QM postulate 1 and the conditions it (The Born Postulate) puts on a proper wavefunction