Lecture 3 Flashcards
3.1. Observables as operators 3.2. The second postulate of quantum mechanics 3.3. The third postulate of quantum mechanics 3.4. The physical significance of eigenfunctions 3.5. The Schrödinger equation in terms of operators
What is the momentum operator?
-ihbarpartial/partial x
What is the energy operator?
ihbarpartial/partial t
What are eigenvalue equations?
Equations in the form:
operator x wavefunction = value x wavefunction
What are eigenfunctions and eigenvalues?
Eigenfunctions are wave functions which obey the eigenvalue equation, and the values in the eigenvalue equation are termed eigenvalues
Do functions that are not eigenfunctions of an operator change shape when operated on?
Yes, for example a cosine function may become a sine function
What is the second postulate of Quantum Mechanics?
“Every dynamical variable may be represented by a Hermatian operator whose eigenvalues represent the possible results of carrying out a measurement of the variable.
Immediately after such a measurement, the wavefunction will be the eigenfunction of the operator, corresponding to the eigenfunction obtained in the measurement.”
Do Hermatian operators have real or imaginary eigenvalues?
Real
What is the third postulate of Quantum Mechanics?
“The operators representing the position and momentum are r and -ihbargrad, respectively.
Operators representing other dynamical variables bare the same functional relation to these (as do the corresponding classical quantities to the classical position and momentum variables.)”
How do you find the expectation value?
Integral over all space of (Psi* A Psi)dx
Where Psi* is the complex conjugate of Psi and A is the operator
How do you calculate standard deviation using expectation values?
sqrt( expectation value of (A^2)) - expectation value of (A), ^2))
What is an expectation value?
A measurement of the mean
What is the Hamiltonian?
The total energy operator. It combines the kinetic and potential energy components.
H^ = -hbar^2/2m grad^2+V
How do you write the 3D time independant Shrodinger equation using the Hamiltonian?
EPsi = H^Psi
What can you deduce from EPsi = H^psi?
The shrodinger equatino is the operator equation that represents energy conservation.
