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[NOTES/QM-10001] Representations in an Inner Product SpaceA brief account of representations in a finite dimensional vector spaces is presented. The use of an ortho norrnal basis along with Dirac notation makes all frequently used formula very intuitive. The formulas for representing a vector by a column vector and an operator by matrices are given. The results for change of o.n. bases are summarized. |
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24-06-22 11:06:31 |
[NOTES/QM-10002] Coordinate Representation$\newcommand{\DD}[2][]{\frac{d^2 #1}{d^2 #2}} \newcommand{\matrixelement}[3]{\langle#1|#2|#3\rangle}\newcommand{\PP}[2][]{\frac{\partial^2 #1}{\partial #2^2}}\newcommand{\dd}[2][]{\frac{d#1}{d#2}} \newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}$ The choice of orthonormal basis of eigenvectors of position operator gives rise to the coordinate representation. The wave function, being the expansion coefficient of state vector in this basis, gives the probability amplitude for outcomes of position measurements.In the coordinate representation the momentum operator assumes a simple form $\widehat{p} =-i\hbar \dd{x}$. |
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24-06-22 11:06:09 |
[NOTES/QM-09001] Unitary Operator for Time Evolution $\newcommand{\DD}[2][]{\frac{d^2 #1}{d^2 #2}}$ That assumption that the superposition principle be preserved under time evolution leads to unitary nature of the them evolution operator. The state vector satisfies differential equation, the Schrodinger equation, with Hamiltonian as the generator of time evolution. |
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24-06-22 09:06:43 |
[NOTES/QM-09002] Time Variation of Average Values$\newcommand{\DD}[2][]{\frac{d^2 #1}{d^2 #2}}$ Assuming time development of states to be given by \[i\hbar \dd[\ket{\psi, t}]{t} = H \ket{\psi t}, \] an equation for time variation of average value of a dynamical variable is derived. Classical correspondence is used to identify the generator of time evolution with Hamiltonian. A dynamical variable not depending explicitly on time is a constant of motion if it commutes with the Hamiltonian. |
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24-06-22 09:06:35 |
[LECS/QM-10] Working with Representations |
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24-06-22 06:06:06 |
[NOTES/QM-10003] A Summary of Coordinate and Momentum Representation $\newcommand{\DD}[2][]{\frac{d^2 #1}{d^2 #2}}\newcommand{\matrixelement}[3] \langle#1|#2|#3\rangle} \newcommand{\PP}[2][]{\frac{\partial^2 #1}{\partial #2^2}}\newcommand{\dd}[2][]{\frac{d#1}{d#2}}\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}$ |
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24-06-22 06:06:37 |
[NOTES/CM-08009] Centrifugal force --- Effect of Earth's RotationThe effect of Coriolis force on the force acting on a body on the earth is computed at the poles, equator and at a general point. he banking of railway tracks and of roads is briefly discussed. |
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24-06-19 16:06:55 |
[NOTES/CM-08015] Rotations in Three DimensionsThe set of all rotations that can be implemented physically in three dimensions form a group. These most important and frequently used rotations are the rotations which can be implemented physically. These rotations, do not change the handedness of the coordinate axes and are called proper rotations. The improper rotations take left handed systems to right handed systems, or vice versa. The statement of Euler's theorem about rotations is given. |
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24-06-18 05:06:55 |
[NOTES/CM-08013] Rotation of a Vector about \(X_3\)- AxisWe derive the transformation rules for a rotation about \(X_3\)- axis. The concept of active and passive rotations is briefly explained.$\newcommand{\Label}[1]{\label{#1}}, \newcommand{\Prime}{^\prime}\newcommand{\eqRef}[1]{\eqref{#1}}$ |
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24-06-17 22:06:43 |
[NOTES/CM-08011] Rotations about a fixed axisRotations about a fixed axis form a one parameter subgroup of rotations. |
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24-06-17 22:06:47 |