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Description
Flashcards by emmalmillar, updated more than 1 year ago
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Created by emmalmillar
over 8 years ago
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| Question | Answer |
| Quantum Mechanics | Quantum Mechanics |
| What is the superposition principle? | "The linear composition of simpler wavefunctions" If |a1> and |a2> are linearly independent states, then |u>=c1|a1>+c2|a2> is also a state of the system. The set of kets|ai> are said to be complete, if we can construct any state |u> from them. |
| What is linearity? | An operator A is said to be linear if: |
| All Physical Observables have corresponding operators, which are: | |
| What is the relationship between the Hamiltonian and the energy? | The Hamiltonian is the operator corresponding to the total energy of the system. |
| Define Expectation | This is used to relate a physical observable to the quantum calculation. The probability of finding a specific state in a mixed eigenstate. The modulus squared is the probability. |
| EigenFunction | "What is measured". When an operator acts on the eigenfunction it gives us the eigenvalue and eigenfunction. |
| Eigenvalue | "The result of the measurement, this is the only physical observable" |
| Eigenstate | "The state at which the operator is exactly the eigenvalue with no uncertainty" |
| Eigenvector | "An eigenfunction is an eigenvector which is also a function" |
| If the wavefunction is not an eigenfunciton of the operator what happens to the measurement? | A measurement of the operator may give different eigenvalues with different probabilities each time. |
| When is an operator said to be Hermitian? | Hermitian operators have real eigenvalues. When it obeys the relationship: |
| Describe Commutation | When 2 operators commute, their observable are simultaneously and precisely observable. |
| Define the Identity Operator | Not all operators have inverses, those that do are called nonsingular. |
| If A|u>=w> What transforms <u| to <w| | This operator is called the adjoint of A, and if equal they are hermitian or self adjoined. |
| Rule One | "The Linear Super Position Principle" Multiplying a vector by a complex number changes the magnitude but not the direction. |
| Rule Two | "An isolated quantum state evolves continuously and casually in time" |
| Rule Three | "Each observable of a system is associated with a Hermitian operator on a Hilbert space of the system. Eigenstates of each observable form a complete set". |
| Rule Four | The result of any given measurement of an observable is an eigenvalue of the corresponding hermitian operator. |
| Rule Five | The expectation of A is the average number of measurements of A. |
| Rule Six | The Probability amplitude <u|a> and the probability is the modulus of this squared. |
| Rule Seven | "The collapse reduction postulate" A wave function—initially in a superposition of several eigenstates—appears to reduce to a single eigenstate by "observation". |
| Which two rules are incompatible? | Rule 2 and Rule 7 |
| When are two observables said to be compatible? | If the operators representing them have a common set of eigenfunctions. If one quantity is measured then the system will be left in an eigenfunction of that observable. A measurement of the other observable will leave the system in the same state. |
| What can be concluded about these two hermitian operators: [A,B]=iC | C is also a hermitian operator. As A and B do not commute they will not have simultaneous eigenstates |
| Ehrenfest's Theorem | The expectation value of any time independent operator is constant if the operator commutes with the Hamiltonian. |
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