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# QUANTUM MECHANICS PHYS 324

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This 7 page Class Notes was uploaded by Dr. Simeon Wiza on Wednesday September 9, 2015. The Class Notes belongs to PHYS 324 at University of Washington taught by Staff in Fall. Since its upload, it has received 16 views. For similar materials see /class/192454/phys-324-university-of-washington in Physics 2 at University of Washington.

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Date Created: 09/09/15

Physics 324 Fall 2002 Dirac Notation These notes were produced by David Kaplan for Phys 324 in Autumn 2001 1 Vectors 11 Inner product Recall from linear algebra we can represent a vector V as a column vector then VJf VT is a row vector and the inner product another name for dot product between two vectors is written as AiBAjB1ABZ 1 In conventional vector notation the above is just A Note that the inner product of a vector with itself is positive de nite we can de ne the norm of a vector to be M V VTV 7 2 which is a non negative real number In conventional vector notation this is which is the length of V 12 Basis vectors We can expand a vector in a set of basis vectors i provided the set is complete which means that the basis vectors span the whole vector space The basis is called orthonormal if they satisfy l j 67 orthonormality 3 and an orthonormal basis is complete if they satisfy 2 El r I completeness 4 where I is the unit matrix note that a column vector times a row vector E is a square matrix following the usual de nition of matrix multiplication Assuming we have a complete orthonormal basis we can write VIVZ VEZV i WE lV 5 The V are complex numbers we say that V are the components of V in the i basis 13 Eigenvectors as basis vectors Sometimes it is convenient to choose as basis vectors the eigenvectors of a particular matrix In quantum mechanics7 measurable quantities correspond to hermitian operators so here we will look at hermitian matrices A hermitian matrix is one satisfying M M E MT hermitian 6 This just means that the components of a hermitian matrix satisfy Mij M2 We say that the vector vn is an eigenvector of the matrix M if it satis es Mun Ann 7 where An is a number called an eigenvalue of M If M is hermitian7 the eigenvalues An are all real7 and the eigenvectors may be taken to be orthonormal 711nm 6mm 8 So we can take the vn to be our basis vectors7 and write an arbitrary vector A in this basis as A 2AM 9 where the An are in general complex numbers This is a convenient choice if we wish to know what is the action of the hermitian matrix M when it multiplies the vector A MA ZAnMvn ZAnAnvn 10 2 Dirac notation for vectors Now let us introduce Dirac notation for vectors We simply rewrite all the equations in the above section in terms of bras and kets We replace V gt m Vt gt ltVl ATB gt ltAlBgt 11 Suppose we have basis vector W analogous to the i which form a complete orthonormal set lt ljgt 617 orthonormality 2139 WW 1 completeness 12 where 1 is the identity operator it has the property 1l1bgt lwgt for any Then any vector lVgt may be expanded in this basis as M 1vgt mew 22 v2 lt lVgt lt13 2 Note that ltVl gt As before we can use the eigenvectors of a hermitian operator for our basis vectors Matrices become operators in this language M a M Then the eigenvalue equation becomes MW MW 7 14 where the An are real and we can take the kets to be orthonormal 6m Then we can write Mm MZanm ZVanm ZVnAnlm 15 3 Dirac notation for quantum mechanics Functions can be considered to be vectors in an in nite dimensional space provided that they are normalizable In quantum mechanics wave functions can be thought of as vectors in this space We will denote a quantum state as This state is normalized if we make it have unit norm ltwlwgt 1 Measurable quantities such as position momentum energy angular momentum spin etc are all associated with operators which can act on lf 0 is an operator corresponding to some measurable quantity its expectation value is given by Note that since A A ltAl0lBgt ltBl0llAgt7 check this in the case Of nite length veetars and matricesf it follows that if your mea surable quantity is real and they always are then implies that WOW Wow or that O Ol Conclusion Measurable quantities are associated with hermitian operators In order to compute expectation values for given quantum states it is often useful to choose a convenient basis I will discuss three common bases that are often used the position eigenstate basis the momentum eigenstate basis and the energy eigenstate basis 31 Position eigenstate basis The position operator i fl is a hermitian operator and we can use its eigenvectors as an orthonormal basis The state is de ned to be the eigenstate of i with eigenvalue x 16 What is new here is that the eigenvalues x are not discrete and so we use the Dirac 6 function for normalization ltxlx gt 625 7 x orthonormality 17 The states form a complete basis for our space and dx 1 completeness 18 Note that the sum over discrete basis vectors in eq 12 has been replaced by an integral Also the unit operator 1 has replaced the unit matrix I Therefore we can expand our state 170 in terms of the basis vectors 1w wmmmwmwmmv m where we have de ned 7x E This 7x is nothing but our familiar wavefunction In the present language 7x are the coordinates of the our state 170 in the basis Note that in eq 19 we inserted the unit operator in the guise of a integral over this technique is very powerful and is called inserting a complete set of states If we have normalized 170 so that 171 1 it follows that 1wwwmwmwmmwmwmwm m This is the usual normalization condition we have seen Computing for our state is particularly easy in the basis since is an eigen state of the operator in awltwmw dxltwli zgtltxlwgt dxxn lgtltl gt dxx 21 In the next section I will discuss measuring 13 using the position eigenstate basis 32 Momentum eigenstate basis p Another useful basis is formed by the eigenstates of the momentum operator 13 mm plpgt 7 129 5p7p 7 dplpgtltpl 1 22 We can expand 170 in the momentum basis as 1wwmwwwamm amzmw am We see that 151 is just the wavefunction in the momentum basis As in the above section we can easily compute expectation values such as 13 using this basis It is interesting to ask how we can translate between the and the lp bases For this7 we need to know the quantity We can get this by knowing that 1x and 151 are Fourier transforms of each other dp W96 27Th 1529 6 h 24 We can rewrite this as dp 27Th Inserting a complete set of states on the left hand side of the above equation we get dpltxlpgtltplwgt lawmanh lt26 wzltzlwgt Mewh lt25 implying that eipwh m 39 Using this result we can also compute here is an eigenvector of i with eigenvalue 1 ltyizaizgt dpltyizaipgtltpizgt dpp ltylpgtltplxgt L ink1 dp 27Th e 395 diam7mer ltlegt 27 2 27139 6 zha miy 28 Therefore if we know 1x but not 1517 we can still compute 13 as ltI3gt WW dy dz ltwlygtltyl lxgtltxlwgt dy mm mgm 7 m we lt29 lntegrating by parts with respect to x ignoring the boundary terms at z ioo7 which vanish we get lt2 dy dwltygt6ltxeygtemwltzgt dwm 421 we 30 So we see that in the x representation 13 a 727337 5 33 Energy eigenstates Finally7 another operator of interest is the Hamiltonian H which gives the energy Up to now7 the Hamiltonians we have seen take the form o2 H 2V2 31 The time dependent Schrodinger equation can be written as a A mama letgt lt32 You can think of Wat as a vector moving around in our vector space as a function of time7 and the above equation governs how it moves Since 13 and i are hermitian7 then H is also hermitian7 provided that the potential Vx is a real function Therefore we can use eigenstates of H as basis vectors HM EM 7 W71 5mm Zlngtltnl 1 33 Note that this eigenvalue equation is simply the time independent Schrodinger equation7 and that since H is hermitian7 the eigenvalues En are real numbers Here I have assumed that the energy eigenvalues En are discrete this is correct for bound states but not scat tering states For states with continuous eigenvalue7 replace the 6m by 6m 7 n7 and the D by f dn Then we can expand Wat in this basis7 with time dependent coef cients Wt 2 CNN 34 Plugging this into the Schrodinger equation eq 32 we nd 5 don t A mama Z zhlngt Z Hentlngt Z mum lt35 Since the form an orthonormal basis7 it is easy to show that the above equation implies z39hdcgiit Encnt gt cnt e iEquotthcn0 36 Therefore the solution for Wat is W t Z e iEMECAONW 37 n So all we need to know is what are cn0 the initial conditions at t 07 and the solutions ln7 En to the time independent Schrodinger equation7 eq 33 Note that lt71 tl ZeiEquotth6n0ltnl 38 6 lf lwtgt is normalized it follows that 1 lt 7tl 7tgt Ct6ntltmlngt WW2 lCn0l2 39 Note that in the coordinate basis where i H x 13 H iihddx ltlngt WM 7 40 where Vx hz d2 41 7 Solving this sort of equation for different potentials Vx and generaliztions in 3 dimen sions what we will be doing for the rest of this quarter 34 Comments Why are these bases we have discussed necessarily different from each other For example canlt we nd a basis in which both 13 and i are simple No If we had a state lx1gt which was simultaneously an eigenstate of i with eigenvalue x and 13 with eigenvalue 1 it would follow that li7 llx7pgt i lx7pgti ilxmgt pilx7pgtim lx7pgt pxixplx7pgt0 42 since x and 13 are ordinary numbers and commute both with each other and with operators 13 and i But we know that 9213 mi 43 we dont normally write the l which is inconsistent with the previous result and so there cannot be such as state as lx13gt The result is that a state lwgt cannot simultaneously be an eigenstate of two operators that do not commute For the hydrogen atom we will nd that we can nd basis sates which are simultaneously eigenstates of H I 13 and IL where L refers to the angular momentum vector We will discuss a normalized basis of angular momentum states later in the course

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