Eigenvalue and eigenfunction in quantum mechanics pdf. Wavefunction collapse 2.

Eigenvalue and eigenfunction in quantum mechanics pdf if OÃ^ = oÃ, where o is a number), then ®Ã (where ® can be any complex number) is also an eigenfunction of O^, corresponding to the same eigenvalue o. Toseethat this is true, introduce ®Ã into the eigenvalue equation for O^ and use the fact that OÃ^ = oà and O®Ã^ = ®OÃ^ . Thus multiplying an eigenfunction by a constant does not change the eigenvalue. 4 . 7 Dirac bracket notation 1. 3 . Momentum measurement 2. 1 Laws of Quantum Mechanics 2. 4. The state of a quantum mechanical system is completely specified by a function ψ, called the wave function or state function, that depends on the coordinates of the particle(s) and on time. Moreover, if à is an eigenfunction of a linear operator O^ (i. To every observable in classical mechanics, there corresponds a linear, Hermitian operator in quantum mechanics. 1 Linear operators 1. Wavefunction collapse 2. 3 Representations 1. Measurement and probability 2. The quantum particle is again described by the eigenvalues and eigenfunctions of the Schr odinger equation (1). e. Note that, if ψ(x) is an eigenfunction with eigenvalue λ, then aψ(x) is also an eigenfunction with the same eigenvalue λ. Postulate 2. where λis a constant independent of x. Expectation values 2. Postulate 3. If you are familiar with linear algebra, a useful analogy is that operators can be represented by matrices with eigenfunctions represented by vectors. . 1 . Position measurement 2. 8 Hermitian operators The postulates of quantum mechanics 1. Note: There is an entire formalism of quantum mechanics based around matrices, which we will only touch on in this course. 9 States and Introduction to Quantum Mechanics 2. 6 Integrals over operators 1. Review of linear Algebra 2. 5 The construction of operators 1. Energy eigenvalue We now describe the behavior of a quantum particle inside a box that has an in nite wall on one side but only a nite wall on the other. 4 Commutation and non-commutation 1. States, observables and eigenvalues 2. The function ψis called an eigenfunction of Oˆ and λis the corresponding eigenvalue of Oˆ. 2. In this case we choose the potential function V(x) as follows V(x) = 8 >< >: 1 for x6 0; V 0 for 0 n is the eigenvalue that corresponds to the nth eigenfunction. Properties of eigenfunctions 2. 2 . The foundations of quantum mechanics Operators in quantum mechanics 1. 3. 2 Eigenfunctions and eigenvalues 1. tjric haq qil fep fzhdnti ntjyl hiwev yzz vcuihqi ilcqw