Chapter 3

quantum state	In position space the state of a quantum system is given by the amplitude for a particle to be found at all possible locations x. This is a complex function .
Probability density	provides a probability to be found in a region where probability is understood in terms of an ensemble of prepared states.
Operators	In position space there are operators that transform functions into other functions.
momentum operator in position space
Hamiltonian
Operators have eigenequations	n signifies that there may be many solutions.
Observable have associated operators	Not all operators are necessarily associated with an observable.
Momentum eigenstates
Energy Eigenstates	Depend on the potential. For V=0è
Postion Eigenstates	delta function
expectation value	Way of characterizing properties of a Quantum State, QS.
Higher moments
Time evolution
Energy Eigenstates	; based on time evolution
Stationary states Energy eigenstaes	Observable energy remains same. Energy eigenstates are stationary i.e. don’t change wrt a measurement of energy.
wave packets; group velocity
general solution	Sum over special solutions !!!!!!!!!

Take the state

A linear combination of two definite momentum states, one particle traveling plus è the other minus ç. This particle has a definite energy but not a definite momentum.

In general find solutions to Schrodinger time dependent equations

The solution of such an equation requires that the left and right hand side be equal for all x and for all t, therefore a constant. Let there be a set of constants that might work and label them with n.

A particular solution to the equation is

where satifies the equation with constant and is a solution to the equation but it is not the most general solution.

The complete solution is

Examine a very special function

This satisfies the time dependent Schrodinger equation with a specific value of the energy. The explicit time dependence tells us that this is a stationary state or an energy eigenstate. The Hamiltonian for this state is not provided. However there must be a confining potential that keeps the particle localized with the x-dependence as shown above. [Notice that on page 199 equ.7.53 shows the ground state for a harmonic oscillator is a Gaussian.]

This is a Gaussian and you should know that

First integral is zero because it is odd. The second is just .

Similarly

You can define new operators

(x-x_o)² ; (p-p_o)²

and find their expectation values.

These are interpreted in the usual sense as the spread.

This is a QS where the particle has a range of momenta and a range of locations. It is spread out in space and momentum-space. Such a state has the minimum for the uncertainty principle.

Comments:

• The expectation value does not need to be a possible measured value. Consider a particle that has a 50% probability to be in the region x=-100 (+/-10) and a 50% possibility of being at x=100 (+/-10). The expectation value is 0 but a measurement will never find the particle at zero. The energy eigenstate with equal amplitudes for è ,ç momentum has an expectation value of p=0.
• Two energy eigenstates can be added together to form a legitimate QS.
• The Gaussian wavefunction above is a contrived state that has a very simple time dependence and Gaussian x dependence. This state needs to have a holding potential. A free particle put into a Gaussian wavepacket will have a different time dependence (Chap 4).
• Separation of variable is a mathematical technique that breaks a many variable differential equation into a product of solutions where each function depends on one variable.
• You should recognize as an eigenequation.
• The general solution is given as a sum.