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# PHYSICAL CHEMISTRY CHEM 455

UW

GPA 3.92

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This 23 page Class Notes was uploaded by Carmela Kilback on Wednesday September 9, 2015. The Class Notes belongs to CHEM 455 at University of Washington taught by Xiaosong Li in Fall. Since its upload, it has received 18 views. For similar materials see /class/192576/chem-455-university-of-washington in Chemistry at University of Washington.

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Date Created: 09/09/15

The Quantum Mechanical Postulate Six postulates Two additional postulates Something about QM postulates QM is based on 6 postulates 5 in current Chapter Postulates can t be p1 uved lelUlty can ue tested by experiments No Violation of the postulates has been found in the 70 years since their formulation QM Postulate 1 Postulate 1 wave function postulate The state of a iuantum mechanical s stem is comi letel specified by Px 1 It All knowable about system contained in P x z Px0t0 2 who to Px0 t0dx Px0t02 dx t Mathematical conditions for a wellbehaved wave function Single valued Continuous T Normalizable quadratically integrable x I P x I 056 i too QM Postulate 1 E Mathematical conditions for a wellbehaved wave function 3 Singlevalued OM Postulate 1 IT Mathematical conditions for a well behaved wave function I Continuous ie m must be finite 6 1 2 x t tanx QM Postulate 2 Postulate 2 operator postulate every observable has a corresponding operator Momentum 4773 13 6x 39 712 62 A 1 A A Kmetlc Energy W Elme c 2m xPx Potential Energy Vx polemm H TotalEnergy a Vx 2711836 QM Postulate 2 Operators 7811111 2m fx21fx1 fx difference 121 1fx121fx1 fx product 211fx211 fx 211fx 1 21fxm Example 2213 i 1322 OM Postulate 3 Postulate 3 observable postulate In any single measurement of the observable that corresponds to the operator the only values that will ever be measured are the eigenvalues of that operator PAILP xt En Pn xt QM Postulate 4 Postulate 4 expectation value postulate The expectation value of the observable a is given by I V x Il l x Ialx a w L P xl l xlalx T Case 1 If Px I is a normalized eigenfunction of 121 l Pxt a l xt a I Pxt1 l xtdx a Px t Px tdx 05 OM Postulate 4 i Case 2 If Px t is not an eigenfunction of 121 Pxt can be expanded in eigenfunctions 0f 21 121 nkxU ang nw lPQCJ an nxt a IlPxat1alPXtdX X21anan n Xtdx 22bbnanj xl nxtdx m1 n1 Zlbmlzam OM Postulate 4 The expectation value is a weighted average of the different possible eigenvalues The measurement process in QM is inherently probabilistic There is no way to predict the outcome of an individual experiment if the wave function is not an eigenfunction of the operator OM Postulate 5 Postulate 5 time evolution postulate The evolution in time of a fuantum mechanical s stem is 7 overned by the time dependent Schrodinger equation 8 x t ELF xl 1 l at H x9 t E x9 t LPx t xe i Eh OM Postulate 6 Chapter 10 Postulate 6 antisymmetry postulate wave functions describing a many electron system must change sign be antisymmetric under the exchange of any two electrons 391 1 81111 EXCIUSIOH pr1n01ple Example The wave function for a system can be written as 4 1 4X 8 lt25 x are normalized eigenfunctions of the total energy Hamiltonian with eigenvalues Enna where a is a constant we 1x 2ltxgt 3x 1 What are the A robabilities of obserVinU E1 E2 E3 and E 4 respectively Example 2 What is the expectation value of the total energy Example 3 Formulate the time dependent wave function for this system Example 4 If half of the particles in state 1 are exited to state 2 what is the new wave function of this system OM Postulate Postulate basis set postulate the set of functions 9 which are eigenfunctions of the eigenvalue equation 19w 2 Elly form a complete set of linearly independent functions f x Zeacul T This implies that any wave function fx representing a physical system can be expressed as a linear combination of the eigenfunctions of any physical observable of the system OM Postulate Complete set F Eivenfunctions of a uantum mechanical o erator form a complete set like any vector in Cartesian coordinate Faib7cZ any wave function fx anyno where 1n x are the eigenfunctions of a QM operator OM Postulate Complete set T Example Fourier series Expand f x in a set of orthogonal functions f x Z Fnem or fx 2 d0 le sinnx dncos n21 OM Postulate Complete set T Example Fourier series x fx OM Postulate Postulate Hermitian property postulate The quantum mechanical operator 3 associated with an observable must be Hermitian Ir 3pr dr or Pbdr

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