{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Chapter3_all

# Chapter3_all - PHY4604 R D Field Eigenfunctions and...

This preview shows pages 1–3. Sign up to view the full content.

PHY4604 R. D. Field Department of Physics Chapter3_1.doc University of Florida Eigenfunctions and Eigenvalues Eigenvalue Equation: Consider the special case of an hermitian operator A op = A op acting on a wave function | ψ> such that A op |ψ> = a |ψ> , where a is a constant. The constant a is called the “eigenvalue” and the wave function | ψ > is called the “eigenfunction” or “eigenket” . Eigenkets have Unit Norm: We require that the eigenkets have finite norm and we normalize the wave functions such that < ψ | ψ > = 1 .. Eigenvalues Value of Hermitian Operators: The eigenvalues values of hermitian operators are real. Proof: We know that < ψ |A op | ψ >=a and taking the complex conjugate of both sides gives a* = < Ψ |A op | Ψ > * = < Ψ |A op | Ψ > = < Ψ |A op | Ψ > = a, Thus, a = a* which implies that a is real. Expectation Value of A op : The expectation value ( i.e. average value) of A op is given by <A> = < ψ |A op | ψ > = a< ψ | ψ > = a . Also, <A 2 > = < ψ |A 2 op | ψ > = a 2 < ψ | ψ > = a 2 . Deviation from the Mean: The square of the deviation of A from the mean, <A> , is given by σ 2 = ( A) 2 <(A - <A>) 2 > = <A 2 > - (<A>) 2 = a 2 –a 2 = 0, and σ = A = 0. This means that the dynamical quantity A op can only have the definite value a ( i.e. precise value). Ordinarily, when you measure an observable O on an ensemble of identically prepared systems, all in the same state | Ψ >, you do not get the same result every time. For “determinate states” you do get the same result every time, you get the eigenvalue. Eigenfunctions of Hermitian Operators with Different Eigenvalues: Now suppose that A op | ψ 1 > = a 1 | ψ 1 > and A op | ψ 2 > = a 2 | ψ 2 > with a 1 a 2 , then < ψ 2 | ψ 1 > = 0 . Eigenfunctions of hermitian operators corresponding to different eigenvalues are orthogonal. Proof: < ψ 2 |A op | ψ 1 > = a 1 < ψ 2 | ψ 1 > and taking the complex conjugate of both sides yields (< ψ 2 | A op | ψ 1 >) * = a 1 * < ψ 2 | ψ 1 > * = a 1 < ψ 1 | ψ 2 > . But (< ψ 2 | A op | ψ 1 >) * = < ψ 1 | A op | ψ 2 > = < ψ 1 | A op | ψ 2 > = a 2 < ψ 1 | ψ 2 > Thus, (a 2 -a 1 )< ψ 1 | ψ 2 > = 0 which implies that < ψ 1 | ψ 2 > = 0 , where I used the fact that a 1 and a 2 are real since A op is hermitian. Called “determinant states”!

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
PHY4604 R. D. Field Department of Physics Chapter3_2.doc University of Florida Obsevables and Vector Spaces Eigenvalue Equation: Consider the case of an hermitian operator A op acting on a set of wave functions, | ψ n >, such that A op | ψ n > = a n | ψ n > where a n is a constant. Note a n is real and < ψ i | ψ j > = δ ij ( i.e. orthonormal set). Completeness: If the ensemble of eigenkets | ψ n > spans the entire space ( i.e. any “ket” of finite norm can be expanded in a series of these “kets”), then they are said to form a complete set and the hermitian operator A op is called an observable . Vector Space (ordinary vectors): In three-dimensional space any vector can be written as a superposition of the three unit vectors, z a y a x a a ˆ ˆ ˆ 3 2 1 + + = r , where a 1 , a 2 , a 3 are real numbers. The three unit vectors form an orthonormal bases. The simplest way to represent a vector is to specify the three constants a i with respect to the basis, = 3 2 1 a a a a r and
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern