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The Mathematics of Quantum Mechanics 3. State vector, Orthogonality, and Scalar Product

The Mathematics of Quantum Mechanics 3. State vector, Orthogonality, and Scalar Product. Measurement in Quantum Mechanics. Measuring is equivalent to breaking the system state down to its basis states:. The basis states are eigenfunctions of a hermitian operator:.

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The Mathematics of Quantum Mechanics 3. State vector, Orthogonality, and Scalar Product

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  1. The Mathematics of Quantum Mechanics 3. State vector, Orthogonality, and Scalar Product

  2. Measurement in Quantum Mechanics Measuring is equivalent to breaking the system state down to its basis states: The basis states are eigenfunctions of a hermitian operator: The values that can be obtained by measuring are the eigenvalues, with the following probability: How can the expansion coefficients gn be calculated, given the wave functions and the expansion basis?

  3. A Scalar Product in Vectors y v1 v2  x A scalar product is an action applied to each pair of vectors: Geometrically this means that the length of one vector is multiplied by its projection on of the other one.

  4. An Orthonormal Base in Vectors y u1 v x u2 If Then And in general:

  5. State Vectors and Scalar Product (Dirac Notation) Each function is denoted by the state vector: <  (q) |  . A scalar product is denoted by < | f>, and fulfills the following conditions:

  6. Scalar Product of Functions The scalar product of functions is calculated by: For example, for a particle on a ring:

  7. Orthonormal Base Theorem: a set of all the eigenfunctions of a hermitian operator constitutes an orthonormal base. Base Orthonormal When the base is orthonormal the expansion coefficients of the function are calculable by means of a scalar product:

  8. Orthonormal Base - a Particle on a Ring Base functions: Orthonormality: Expansion Coefficients (Fourier Theorem)

  9. Orthonormal Base - Legendre Plynomials The definition space of the functions is on x axis, in the [1,1-]: (x)| The base functions: Orthonormality: Expansion Coefficients:

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