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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

... Part – C (4 x 12.5 = 50 Marks ) (Answer any four questions) 16. Obtain Newton’s second law of motion from Ehrenfest’s theorem. 17. Find the transmission coefficient of a particle moving along the x-axis encountering a potential barrier of breadth ‘a’ and height V0, if the energy of the particle E < ...
763313A QUANTUM MECHANICS II Exercise 1 1. Let A and B be
763313A QUANTUM MECHANICS II Exercise 1 1. Let A and B be

... 3. Calculate the expectation values hα|σi |αi, i = 1, 2, for Pauli spin matrices σ1 and σ2 with respect to an arbitrary state |αi = α1 |1i + α2 |2i. Here |1i and |2i are the eigenvectors of σ3 . 4. Consider a three-dimensional vector space spanned by an orthonormal basis |1i, |2i, |3i. Kets |αi and ...
3.1 Linear Algebra Vector spaces
3.1 Linear Algebra Vector spaces

... hα | T̂ | βi = hα | iihi | T̂ | jihj | βi = αi∗Tij βj = (αiTij ∗)∗βj = (αi{T†}ji)∗βj = ({T†}jiαi)∗βj i.e., T̂ † is that transformation which, when applied to the first member of an inner product, gives the same result as if T̂ itself had been applied to the second vector. Four properties: 1. any Her ...
quantum and stat approach
quantum and stat approach

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

Document
Document

7 - Physics at Oregon State University
7 - Physics at Oregon State University

Homework Set 3
Homework Set 3

... the form of the whole series (leading to sines and cosines); that is, you are not required to do a rigorous demonstration by mathematical induction. As for part 2, you should make explicit use of part 1! Note that the series terminates after the ...
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Commutative Operators and Common Basis
Commutative Operators and Common Basis

< 1 ... 34 35 36 37 38

Compact operator on Hilbert space

In functional analysis, compact operators on Hilbert spaces are a direct extension of matrices: in the Hilbert spaces, they are precisely the closure of finite-rank operators in the uniform operator topology. As such, results from matrix theory can sometimes be extended to compact operators using similar arguments. In contrast, the study of general operators on infinite-dimensional spaces often requires a genuinely different approach.For example, the spectral theory of compact operators on Banach spaces takes a form that is very similar to the Jordan canonical form of matrices. In the context of Hilbert spaces, a square matrix is unitarily diagonalizable if and only if it is normal. A corresponding result holds for normal compact operators on Hilbert spaces. (More generally, the compactness assumption can be dropped. But, as stated above, the techniques used are less routine.)This article will discuss a few results for compact operators on Hilbert space, starting with general properties before considering subclasses of compact operators.
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