Quantum Mech Flashcards
Heisenberg’s Uncertainty Principle
The product of the uncertainties in position and momentum must always be greater than the lower bound h’/2
Complementary variables
Increasing the precision to which we know one variable will reduce the maximum amount of information that is possible to know about the other. (Energy and time, position and momentum)
Born interpretation
The wave function is a probability amplitude. Particle is more likely to be found in regions where the wavefunction has a larger amplitude.
Postulate 1: state
At any point in time, the state of a quantum mechanical system is completely specified by ket(wave-function)
Observable
Any measurable quantity
Expectation value
The average value obtained when an observable is repeatedly measured
Postulate 2: operators
Any observable A has an associated linear operator A^
Postulate 4: Expectation
The expectation value of an observable is given by applying its operator to the wavefunction and multiplying by the complex conjugate of the wavefunction
Postulate 5: Evolution
The wavefunction of a system evolves in time according the time-dependent Schrödinger equation
Boundary conditions:
-Finite
-Single-Valued
-Continuous
-Continuous 1st derivative
Finite
Wavefunction cannot go to infinity as it must be normalisable. It also cannot be 0 everywhere as the particle has to be somewhere.
Single valued
Probability density and wavefunction for the particle must have one value at each position
Continuous
Wavefunction must be continuous so first derivative remains finite (no step change)
Continuous 1st derivative
First derivative must be continuous so second derivative remains finite. (No step changes in first derivative). Unless there is an infinite discontinuity in potential at the boundary.
Stationary states
When the wavefunction’s probability density is constant in time
Zero point energy
Ground state energy where n=0
Parity
The symmetry of an eigenfunction
Odd n
Even parity
Even n
Odd parity
Collapsing the wavefunction
Before measurement, the energy of the particle is not well defined. After measurement, the particle must exist in a definite eigenstate, wavefunction 1 or 2, as it has energy E1 or E2. Hence the act of measurement alters the state of the particlw.
Postulate 3: collapse
In any single measurement of an observable a with an operator A^, the only values that will ever be observed are its eigenvalues a satisfying A^|wavefunction>= a|wavefunction>
Probability flux
The probability per unit time that a particle will be found passing a certain point in a given direction
Quantum tunneling transmission equation assumption
The barrier is wide compared to the rate of exponential decay through the barrier
