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MIT 3 23 - The Hamiltonian of a free particle

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Lecture #3 Today’s Program: 1. Review of previous lecture 2. QM free particle and particle in a box. 3. Principle of spectral decomposition. 4. Fourth Postulate Math tools covered today 1. Learn how to solve separable differential equations. 2. A basis of functions, how to expand a function in terms of a set of basis functions 3. Inner products (dot product in vectors) in function space. 4. A basis of functions, how to expand a function in terms of a set of basis functions 5. The connection between symmetries in the physical system, invariance of the Hamiltonian and conserved physical quantities (this is one of the underlying themes in this course to be expanded on in future lectures) Questions you will by able to answer by the end of today’s lecture 1. How to find statevectors (wavefunctions) for two simple yet important cases. 2. Learn how to verify that a given statevector is an eigenvector (eigenfunction) of an operator. 3. Find the spatial and temporal solutions of Schroedinger’s equation for a system which is described by a time independent Hamiltonian (no explicit time dependence of the Hamiltonian). 4. Show that the spatial solution is in fact an eigenfunction of the Hamiltonian. 5. Show how spatial confinement of the system leads to energy discretization. 6. Know the qualitative differences between the free particle case and the particle in a box. 7. What is the condition that leads to energy discretization? 8. How to find the projection of a function (vector) onto an eigenfunction (element of the basis) 9. How to expand an arbitrary wavefunction in terms of eigenfunctions? 10. How to predict the probability of obtaining a particular measurement result.First Example: The Hamiltonian of a free particle: 22ˆ22classicalpPHHmm=→=rr 2ˆˆˆˆˆˆˆ2PPHxyzxyzmiixyzxyz⋅∂∂∂∂∂∂==∇⋅∇=−++⋅++∂∂∂∂∂∂rrrrhhh focus on a 1D problem for simplicity: To find the eigenfunctions (),uxt of a particular operator one needs to solve the following equation: ()()ˆ,,Auxtuxtλ= The eigenvalues of the Hamiltonian operator are called the energy let us now find an eigenfunction for the operator which we have called the Hamiltonian. Note: we will use the symbol u for a eigenfunction. ()()ˆ,,HuxtEuxt= ()( )()( )( )2222222222202mmiExiExEuxuxmEuxEuxmxxuxaebe−∂∂−=→+=∂∂=+hhhh So we have now the energy eigenfunctions and energy eigenvalues for the free particle case. Please note that these functions are oscillating and spread over all space and that E can assume any value positive or equal to zero. The sixth postulate allows us to find the time evolution of a state using the Schrodinger equation. ()()222,,2xtxtimxtψψ∂∂−=∂∂hh This equation is a partial differential equation, which can be solved using a separation of variables method: ()()(),xtxtψφξ= substituting in the above equation gives, ( )()( )()222112dxdtiEmxdxtdtφξφξ−==hhThe time dependent part becomes: ()( ) ( )0EitdtEittedtξξξ−+=→=hh The spatial part becomes, ()( ) ( ) ( )2222202ikxikxExmExxaebeuxxmkEφφφ+−∂+=→=+=∂=hh Each distinct E corresponds to a different valid solution which has the form: ( )( ),EitikxikxExtaebeeψ−+−=+h so the complete solution can be written as a superposition of the individual solutions: ( )( )1,iiiENitikxikxiiixtaebeeψ−+−==+∑hSecond example: particle in an infinite potential well: I] The system: A particle of mass 2m in a potential well: II] The classical energy function of the system: ( ) ( )2,2pHxpVxm=+ where the potential energy is defined as follows ( )022 22ddxVxddxorx−<<=∞<−> III] Obtaining the QM Hamiltonian operator: ( )( ) ( )( )2222ˆˆˆˆˆˆ,,22PHxpHXPVXVxmmx∂→=+=−+∂h where, ( )022 22ddxVxddxorx−<<=∞<−>. E dFor the moment let us restrict our attention to the region of space which has a finite potential and consider the possible eigenfunctions and eigenvalues of the operator: ( )222ˆˆˆ,2HXPmx∂=−∂h IV] What are the energy eigenfunctions and eigenvalues? ( )( ) ( ) ( ) ( )222ˆˆˆ,2HXPuxEuxuxEuxmx∂=→−=∂h ()( )()( )( )2222222222202mmiExiExEuxuxmEuxEuxmxxuxaebe−∂∂−=→+=∂∂=+hhhh Note: The undetermined a and b coefficients imply that there are an inifinite number of allowed eigenfunctions corresponding to every eigenvalue (i.e. determining E does not determine a and b). We will narrow down this set by using boundary conditions derived from physical insights into our problem. Specifically, we will require that our eigenfunctions will be equal to 0 at the boundary of the well. At the moment we do not have a solid justification for this other than the definition of the problem which is such that the wavefunctions need to be equal to zero at the boundaries and therefore we want to choose a basis set which can be conviently used to express the wavefunctions. The problem of finding the eigenfunctions and eigenvalues of a linear quantum mechanical operator basically is identical to solving a linear differential equation. As such we can apply the techniques and theories which have been developed to solve differential equations to our problems. A differential equation of the type we are considering will have a unique solution provided that the values of the solution are known at the boundaries (boundary conditions).The boundary conditions in this problem are: 02dux=±= ( ) ( )220cossincossin022222cossin022 n=1,3,5 ....22or n=2,4,6 ....22ddikikdkdkdkdkduaebeaibikdkdabababkdnabkdnππ+−=+=→++−=→++−=→=→==−= The form of the solution is, ( ) ( )2 oddcoscos evensinsin2nEnnnnnxnckxcduxuxnxndkxddmE nkdπππ=≡====h ( )( ) ( )( )( ) ( )22222ˆˆˆ,coscoscos22ˆˆˆ,nnnnnnnnnkHXPxakxakxEakxExmxmHXPuxEuxψψ∂=−===∂=hh Because H is a linear operator any superposition of solutions is also a solution. The boundary conditions have lead to a quantization of the energy levels:2222 n=1,2,3,4,5....221=28nkdnnnhEmdmdππ=→=h V] Finding Schrodinger’s equation and the wave function: ( )( ) ( ) ( ) ( )222ˆˆˆ,,,,,2HXPxtixtxtixttmxtψψψψ∂∂∂=→−=∂∂∂hhh We have already solved this equation and the solutions are: ( )( ),Eitikxikxxtaebeeψ−+−=+h 22mEk =h Let us focus on the spatial


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MIT 3 23 - The Hamiltonian of a free particle

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