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# note03 - ECE 3060 Introduction to Quantum and Statistical...

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1 ECE 3060: Introduction to Quantum and Statistical Mechanics Handout 3 Operators, Expectation Values, Eigenvalues and Eigenfunctions Further Reading: Senturia, Chaps. 5 and 6 (pp. 65-110). 3.1 Physical quantities as operators in quantum mechanics We have learned the fundamental postulates in quantum mechanics in the last Chapter. These postulates, though reasonable from the wave-particle duality and asymptotically mapped into the classical conservation laws of momentum and energy, cannot be proven directly. However, we will just accept the postulates now as is, and complete the mathematical formalism of quantum mechanics. Up to now, the observable operators we know include position x , time t , momentum x i p = h ˆ , total energy t i E = h ˆ , the kinetic energy 2 2 2 2 2 ˆ ˆ = = m m p K h , the potential energy V(x) , and the Hamiltonian ) ( 2 ˆ 2 2 2 x V x m H + = h . There are other operators we will introduce later in the semester, some are observables such as angular momentum, but some are non-observables such as damping. If we have confusion whether we are denoting a number or an operator, we often (but not always) will add a head symbol for the operator. For example, = x x ˆ will read as “the position operator x ˆ is defined as the variable x times the operand”. Please also notice that when we write down the operator equations, say, ψ E ψ H ˆ ˆ = , we do NOT imply that E H ˆ ˆ = at all, but just that the results AFTER the operation on ψ is equal. For the Schrodinger equation of ψ E ψ H ˆ ˆ = , we can see that for V(x)=0, if is a plane wave function of e i(kx - ω t) and, m k k ω 2 / ) ( 2 h = then the equality ψ E ψ H ˆ ˆ = is true. The extension can be made to the eigenvalue equation where one of the operators is simply a constant c . If for a specific operator Q ˆ , so that ψ a ψ Q = ˆ , we will call a is the eigenvalue with the corresponding eigenfunction . You have seen the importance of the eigenvalue and eigenfunction in the basic postulates in the previous Chapter. (Exercise 3.1) When we state ψ p ψ p = ˆ with x i p = / ˆ h , what do we mean? (Exercise 3.2) What is the difference when we state ψ E ψ H ˆ ˆ = and ψ E ψ H = ˆ , when E ˆ is an operator defined by t i E = h ˆ and E is a real number? 3.2 Duality of operators in the real and reciprocal space Before we start more formal introductions on the operators in quantum mechanics, we will first clear up the concepts for operators in the real space x and reciprocal space k . This will

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2 be convenient later, since some operators are easier to calculate in one of the spaces than the other. We will use the momentum operators as an example. For a wave packet with ψ (x,t) expressed as the superposition of the plane-wave function for V(x) = 0 : = π dk e k A t x ψ t ω kx i 2 ) ( ) , ( ) ( ( 3 . 1 ) where A(k) is the Fourier transform of the wave function at
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