Final Exam Review!!! Flashcards
study for the final!!!
For a free particle, what is the potential energy function?
Zero
Critical physical difference between a free particle and a bound particle
There is no confining potential for a free particle
For a free particle, is the energy discrete or continuous?
Continuous because there are not enough constraints
Consider the time evolution of the energy eigenstates for a free particle. What do you get and what does it look like?
You get a form that represents a wave that retains its shape as it moves, any given point on that shape moves with a speed determined by the parameter v. Has the same form that we know from classical waves
Phase velocity (free particle state)
The speed at which points of constant phase move
Whenever we see a wave function with spatial dependence exp(p/m kx), the sign of the wave vector in the exponent indicates
The direction of motion
what is the momentum eigenvalue for a momentum eigenstate (free particle)
p = h/lambda
For the free particle, the eigenstates of the momentum are also eigenstates of the
energy
Commuting observables share
Common eigenstates
True or false: for a free particle, a given momentum eigenstate has a definite energy
True
True or false: a given energy state has a definite momentum
False
For a free particle, a general energy eigenstate is a superposition of
Two momentum states with opposite momenta
For a free particle, we say that the energy state is degenerate with respect to momentum because
A given energy state corresponds to multiple momentum states
In the free particle case, the energy states are
Two-fold degenerate
The proper way to use a wave to describe a particle is using the
Group velocity of a wave packet
For a free particle, mathematically, we cannot normalize the momentum eigenstates because
The integral of the probability density over all space is infinite
How do we get wave functions that are normalizable and localized to finite regions of space for the free particle?
We can construct superpositions of momentum eigenstates to make wave packets
What is the continuous analog of the Kronecker delta?
Dirac delta function
The wave function and the momentum space wave function are both representations of
The same state, but they are representing that state in different bases
For a free particle, the state psi can be written in two different bases. What are they?
The position basis using the wavefunction OR the momentum basis using the momentum space wave function
What is the connection between the momentum space wave function and the position space wave function?
The Fourier transform
Why is a wave packet called a wave packet?
Because it is localized to a region of space
What does a wave packet have in common with a wave? With a particle?
It has a characteristic wavelength determined by the central momentum
it has a limited spatial extent
If you have a discrete Fourier sum of momentum eigenstates vs one with a continuum distribution, how are the results different?
For the discrete case, the localization is repeated periodically. For the continuum distribution, only one localized region exists
The envelope of the wave packet moves at a different velocity from the carrier and this velocity is known as the
group velocity
The group velocity
characterizes the velocity of the group of waves together
The Hamiltonian operator corresponds to
The total energy of the system
Which operator determines the time evolution of a quantum system?
The Hamiltonian
The Schrodinger equation is a
Differential equation that tells you the time evolution of a quantum system
What is the energy basis?
The basis of eigenvectors of the Hamiltonian
Why do the eigenvectors of the Hamiltonian form a complete basis?
Because the Hamiltonian is an observable and a Hermitian operator
How do you find the time dependence of an original state vector?
You multiply each energy eigenstate coefficient by its own phase factor that depends on the energy of that eigenstate
What are stationary states?
States that do not evolve with time
True or false: if a system begins in an energy eigenstate, then it remains in that state
True
Consider an operator A. It commutes with the Hamiltonian. What can you conclude about the probability of measuring any particular eigenvalue of A?
It is time independent
If an operator A does not commute with the Hamiltonian, then what can you say in general about the eigenstates of A?
They are superpositions of energy eigenstates
Time dependence is determined by
The difference of the energies of the two states involved in the superposition
If a state is unbound, then is the energy spectrum discrete or continuous and why?
Continuous because all energies above the energy required to escape the well are allowed
If a state is bound, then is the energy spectrum discrete or continuous?
Discrete
What does local curvature tell you?
It tells you what the shape of the wavefunction is
The second derivative of a wavefunction is proportional to the wavefunction, what does this tell you about the wavefunction?
It’s exponential
The second derivative of a wavefunction is proportional to the negative of itself, what does this tell you about the shape of the wavefunction in this region?
It’s oscillatory
Suppose you have a Pauli matrix and you multiply it by some value, how are the eigenvectors affected? How are the eigenvalues affected?
The eigenvectors are the same but the eigenvalues are all multiplied by the constant
For any spin projection (i.e. something like a superposition of S_z and S_x), what are the eigenvalues?
plus or minus hbar/2
If you have something that is a superposition of S_z and S_x but it’s not normalized, then how do you find the eigenvalues?
Recall that the eigenvalues for any spin projection are plus or minus hbar/2, to find the eigenvalues then you just need to find the normalized version and the constant that the normalized version differs from the actual version by
What does the quantum number l called and what does it represent
It is the orbital angular momentum quantum number and it gives a measure of the size of the angular momentum vector in that direction.
What is the magnitude of the angular momentum vector
sqrt(l(l+1)/hbar)
Crucial difference between spin angular momentum and orbital angular momentum (e.g. the allowed quantized values for the spin angular momentum number)
spin case: the allowed quantized values of the spin angular momentum number s are the integers and half integers
orbital case: the quantum number l can only take on integer values
for each l, what are the possible values of m?
Integers from -l to l.
What is the spherical coordinate representation of the angular momentum operator L_z?
-ihbar(d/d(phi))
Does the Hamiltonian commute with the orbital angular momentum operator L^2?
Yes
Does the Hamiltonian commute with the orbital angular momentum operator L_z?
Yes
What properties does a system have with no radial or polar angle dependence?
A particle confined to move on a ring with constant radius
What are the energy eigenstates for the particle on a ring?
The states |m> that satisfy the L_z eigenvalue equation
What are the allowed values for a particle on a ring?
E_m = m^2(hbar^2/2I)
What is the position representation of the |m> state?
1/sqrt(2pi)exp(im(phi))
true or false: |m> is an eigenstate of the ring Hamiltonian
true
true or false: |m> is an eigenstate of the z-component of the orbital angular momentum
true
