02-25-11_Notes

# 02-25-11_Notes - Observables and Eigenstates QM I...

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Observables and Eigenstates QM I Substitute Lecture Notes Chris Mueller Dept. of Physics, University of Florida Last Updated: February 24, 2011 Contents 1 Introduction 1 1.1 Analogy to Vectors in R 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Stationary States 2 3 Discrete Eigenvalues (Bound States) 3 4 Continuous Eigenvalues 3 4.1 The Position Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 4.1.1 Delta Function Orthonormality . . . . . . . . . . . . . . . . . . . . . . . . . 4 4.1.2 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 4.2 The Momentum Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 4.2.1 Aside: The Delta Function in the Fourier Domain . . . . . . . . . . . . . . 5 4.2.2 Delta Function Orthonormality . . . . . . . . . . . . . . . . . . . . . . . . . 5 4.2.3 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 5 Conclusion 6 5.1 Discrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 5.2 Continuous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1 Introduction We will be discussing Hermitian operators, eigenvectors (eigenstates), and eigenvalues and their relation to physical systems. During the discussion and throughout one’s study of quantum me- chanics, it is helpful to keep the following analogy in mind. 1.1 Analogy to Vectors in R 3 Vectors in 3 dimensions ( R 3 ) are something which we are very familiar with at this stage in our studies. Quantum mechanical observables are simply vectors in a diﬀerent space known as Hilbert space. Lets think about the properties of a vector in R 3 . The vector itself is an abstract concept (an arrow) which lives in 3 dimensions and describes some physical quantity. In order to work with the vector we import it into the land of mathematics by expressing it in terms of a coordinate system (an orthonormal basis), ~v = v x ˆ x + v y ˆ y + v z ˆ z = ( ~v · ˆ x x + ( ~v · ˆ y y + ( ~v · ˆ z z, (1) for instance. 1

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The vector is expressed in terms of the coordinate system (orthonormal basis) as scalers multiplied by special vectors which also live in R 3 . Some coordinate systems (orthonormal bases) are much better for solving problems than others. Wavefunctions in Hilbert space work in much the same way, even though they are more alien to us because of our lack of experience. The wavefunction is an abstract concept whose expectation value with an observable operator describes some physical quantity. In order to work with the wavefunction we resolve it into an orthonormal basis such as the eigenvectors of a Hermitian operator ˆ Ω. | Ψ
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## This note was uploaded on 10/26/2011 for the course PHY 2049 taught by Professor Any during the Fall '08 term at University of Florida.

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02-25-11_Notes - Observables and Eigenstates QM I...

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