[QMSE] W04 - Time Dependent Schrodinger’s Equations

QM (quantum mechanics)) explanations have to hold for CM (classical mechanics) phenomena as well
when using the time independent SE(schrodinger’s equation), even at higher eigen-energy states
- oscillation was not explained
- spatial probability distribution was fixed in the eigen-energy wave solutions
- until now only energy eigen-states have been explored
eigen-states have a property
- nothing measurable in eigen states changes over time
systems that change in time need to explored beyond the idea of energy eigen states
- however, this has to be consistent with the time-independent results
a time dependent rationalization has to be worked out
- however, this is not like CM wave equation mathematically
- but describes time-dependent variable in quantum mechanics
this exploration exposes the concept of superposition in QM
- schrodinger’s cat example
- concept of superposition is important to understand time evolution in QM

rationalizing the time-dependent schrodinger’s equations

time-dependent SE is the relation between frequency and energy in QM
- example: electron magnetic waves and photons

photon frequency-energy experiment

consider two experiments with monochromatic light
- a single frequency or single color EM (electro-magnetic wave) is used
in first experiment
- the frequency of the oscillations in the wave is measured
- the number of photons per second corresponding to a particular part of this frequency is counted
in the second experiment
- the number of photons per second on a detector is measured
- the number of photons arriving per second is used to deduce the the energy of photons
  - so the power arriving is measured
the comparison of the results of the experiment results in
- the relationship between energy per photon with the frequency
- which is \( E = h\nu \)
- where:
  - \( h \): planck’s constant
  - \( \nu \): photon frequency
for photons:
- energy is proportional to frequency
- planck’s constant being the constant of proportionality

matter energy transitions

hydrogen atoms, for instance, emit photons as they transition between energy levels
so some oscillation is expected in the electrons at the corresponding frequency during the emission of the photon
- consequently, there is an expectation for a similar relation between energy and frequency associated with electron levels
the proposed frequency-energy QM relationship for a wave is: \[ E = h\nu \]
- \( h \): planck’s constant
- \( \nu \): some frequency yet to be determined
equivalently \[ E = \hbar\omega \]
- \( \omega \): corresponding angular constant
- \( \hbar \): planck’s constant for wavelength relationship

time dependent equation

consider particle with mass \(m\)
- with frequency-energy equation \( E = hv = \hbar\omega \)
assuming plane wave solutions of the form
- \( \exp{\left[ i (kz -\omega t) \right]} \)
- for some specific energy \(E\)
- and a uniform potential

schrodinger’s time-dependent equation

a time-dependent wave equation postulated by Sch. \[ -\frac{\hbar^2}{2m} \nabla^2 \Psi(\textbf{r},t) + V(\textbf{r},t) \Psi(\textbf{r},t) = ih\frac{\partial \Psi(\textbf{r},t)}{\partial t} \]
note that for a uniform potential
- example: \( V = 0 \) for simplicity,
  - with \( E = \hbar\omega \)
  - \(k = \sqrt{\frac{2mE}{\hbar^2}} \)
- waves of the form \[ \begin{align} \exp{\left[ -i(\omega t \pm kz) \right]} & \equiv \exp{\left[ -i \left( \frac{Et}{\hbar} \pm kz \right) \right]} \\
  & \equiv \exp{\left( -i \frac{Et}{\hbar} \right)} \exp{(\pm ikz)} \\
  \end{align} \]
  - are solutions

sign convention

schrodinger chose a sign for the right hand side
- which means that a wave with a spatial part \( \propto \exp{(ikz)} \)
- is definitely going in the positive \(z\) direction
including its time dependence, that wave would be of the form
- \( \exp{\left[ i (kz-\frac{Et}{\hbar}) \right]} \)
- for \(V = 0 \)

compatibility with time-INdependent equation

schrodinger’s time-independent equation (STIE) could apply
- if we had states of definite energy \(E\) (an eigen-energy)
consider a corresponding eigenfunction \( \psi(\textbf{r}) \) for the eigen energy \(E\) \[ -\frac{\hbar^2}{2m} \nabla^2\psi(\textbf{r}) + V(\textbf{r})\psi(\textbf{r}) = E\psi(\textbf{r}) \]
however, this solution \( \psi(\textbf{r}) \) is not a solution of the time-dependent equation \[ -\frac{\hbar^2}{2m} \nabla^2 \Psi(\textbf{r},t) + V(\textbf{r},t) \Psi(\textbf{r},t) = ih\frac{\partial \Psi(\textbf{r},t)}{\partial t} \]
\(\psi(\textbf{r})\) is here for \( \Psi(\textbf{r},t) \) does not work because
- \( \psi(\textbf{r}) \) has no time-dependence
- the right hand side become zero for a non time-independent wave function
  - as the partial w.r.t time is zero for a time independent function
  - when it should result in \( E\psi(\textbf{r}) \)
instead of using solution \( \psi(\textbf{r}) \), use \[ \begin{align} \Psi(\textbf{r},t) & = \psi(\textbf{r}) \exp\left( -i \frac{Et}{\hbar}\right) \\
-\frac{h^2}{2m} \nabla^2 \Psi(\textbf{r},t) + V(\textbf{r}) \Psi(\textbf{r},t) & = - \frac{\hbar^2}{2m} \nabla^2 \psi(\textbf{r}) \exp{\left( -i\frac{Et}{\hbar} \right)} + V(\textbf{r}) \psi(\textbf{r}) \exp{\left( -i \frac{Et}{\hbar} \right)} \\\ & = \left[ - \frac{\hbar^2}{2m} \nabla^2 \psi(\textbf{r}) + V(\textbf{r}) \psi(\textbf{r}) \right] \exp{\left( -i\frac{Et}{\hbar} \right) } \\
& = E\psi(\textbf{r})\exp{ \left( -i \frac{Et}{\hbar} \right)} \\
& = E\Psi(\textbf{r},t) \\
\end{align} \]
so the expression \[ \Psi(\textbf{r},t) = \psi(\textbf{r}) \exp{\left(-i\frac{Et}{\hbar}\right)} \]
- solves the time-dependent schrodinger equation
similarly, knowing that \( \psi(\textbf{r}) \)
- solves the time-INdependent equation with energy \(E\)
- substituting \( \Psi(\textbf{r},t) = \psi(\textbf{r}) \exp{ \left( -i\frac{Et}{h} \right) } \)
  - in the ime-dependent equation gives \[ \begin{align} -\frac{\hbar^2}{2m} \nabla^2 \Psi(\textbf{r},t) + V(\textbf{r}) \Psi(\textbf{r},t) & = i\hbar \frac{ \partial \Psi( \textbf{r},t ) }{\partial t} \\
    & = i \hbar \frac{\partial}{\partial t} \left[ \psi(\textbf{r})\exp\left(-i \frac{Et}{\hbar} \right) \right] \\
    & = i \hbar \psi(\textbf{r}) \frac{\partial}{\partial t} \left[ \exp{\left (-i \frac{Et}{\hbar} \right)} \right] \\
    & = i\hbar \psi(\textbf{r})\left[ -i\frac{E}{\hbar} \right] \exp\left(-i \frac{Et}{\hbar}\right) \\
    & = E\Psi(\textbf{r},t) \end{align} \]
- so \( \Psi(\textbf{r},t) = \psi(\textbf{r})\exp{ \left( -i \frac{Et}{\hbar} \right) } \)
  - solves the time-dependent schrodinger equation
  - as long as we always multiply it by a factor \( \exp{ \left( -i \frac{Et}{h} \right) }\)

time-dependent from time-INdependent solutions

if \( \psi(\textbf{r}) \) is a solution of the time-independent schrodinger equation,
- with eigen-energy \(E\)
then \( \Psi(\textbf{r},t) = \psi(\textbf{r}) \exp{\left( -i \frac{Et}{\hbar} \right)}\)
- is a solution of both the time-dependent and time-independent SE
- making these two equations compatible

oscillations and time-independence

if the following is a proposed solution to a time-independent problem \[ \Psi(\textbf{r},t) = \psi(\textbf{r}) \exp{\left( -i \frac{Et}{\hbar} \right)} \]
- this maybe used to represent something stable in time
- measurable quantities associated with this state are stable in time

example:

probability density = square of modulus of wave function
square of modulus of wave function:
- wave function conjugate multiplied with wave function \[ \begin{align} \lvert \Psi(\textbf{r},t) \rvert^2 & = \left[ \exp{\left( i\frac{Et}{\hbar} \right)} \psi^*(\textbf{r}) \right] \times \left[ \exp{\left(-i \frac{Et}{\hbar} \right) }\psi(\textbf{r})\right] \\
  & = \lvert \psi(\textbf{r}) \rvert^2 \\
  \end{align} \]

notes

time-dependent SE is not an eigen value equation in general
- i.e. not an equation that only has solutions for a particular values of a parameter
  - parameter is not the eigen value
  - evaluation of the wave equation with a particular value of the parameter results in an eigen value
supports a much richer set of solution

solutions of the time-dependent SE

common classical wave equation \[ \nabla^2 f= \frac{k^2}{\omega^2} \frac{\partial^2 f}{\partial t^2} \]
for which \[ f \propto \exp{\left[ i(kz - \omega t) \right]} \]
- would also be a solution
classical wave equation is not complex wave
classical equation has a time derivative
- as opposed to the first time derivative
- in Schrodinger’s time-dependent equation (STDE)
STDE is a complex wave equation \[ -\frac{h^2}{2m} \nabla^2 \Psi(\textbf{r},t) + V(\textbf{r},t)\Psi(\textbf{r},t) = i\hbar \frac{\partial \Psi(\textbf{r},t)}{\partial t}
\]
- \( \Psi \) is required to be a complex entity
for example \(V = 0\)
- though \(\exp[i(kz - \frac{Et}{h})]\) is a solution
- \( \sin\left( kz-\frac{Et}\hbar{} \right) \) is not a solution
while in CM, complex waves are converted to real waves
- by adding conjugate waves
this is not done in QM, because complex waves are dealt with directly
this because of two reasons
1. the wave function is not a physical wave, it does not have any meaning by itself
  - it is a mathematical tool to calculate behavior
  - it cannot be measured physically
  - so it’s complexity does not matter
2. to extract measurable quantities from calculations
  - the modulus squared of the wave function is used
  - more generally, the QM amplitude is used
  - this gets us to the real world
in CM, knowledge of the wave at a given moment does not help predict the future of the wave
- additional information like the time derivative is needed for this
the QM wave function, just from the current form of the wave
- what happens next is already known
- the wave function embodies everything we can know about the QM state and consequently its future
- this is the justification for moving from the real waves of CM to complex waves in QM

time evolution form SC

consider STDE \[ -\frac{\hbar^2}{2m} \nabla^2 \Psi(\textbf{r},t) + V(\textbf{r},t)\Psi(\textbf{r},t) = ih \frac{\partial \Psi(\textbf{r},t)}{\partial t} \]
if we knew the wavefunction \( \Psi(\textbf{r},t) \) at every point in space at some time \(t_0 \)
- the LHS of the STDE can be computed at that time for all \( \textbf{r} \)
- so we would know \( \frac{\partial \Psi(\textbf{r},t)}{\partial t} \) for all \( \textbf{r} \)
- so we could integrate the equation to deduce \( \Psi(\textbf{r},t) \) at all future time
explicitly, knowing \( \left( \frac{\partial \Psi(\textbf{r},t)}{\partial t} \right) \)
- calculate a future wave; for instance, a small instance \( \delta t \) later in time \[ \Psi(\textbf{r}, t_0 + \delta t) \cong \Psi(\textbf{r},t_0) + \frac{\partial \Psi}{\partial t} \Bigg\vert_{\textbf{r},t_0} \delta t \]
- that is, we can know the wave function in space at the next instant in time
- can be continued infinitely to predict all future evolution of the wavefunction
  - this is why the STDE has first time derivative component
  - as opposed to the second time derivate component
  - knowing only the second time derivative cannot be used to find the evolution in time
    - this means both a future and a past time
any spatial function could be a solution of the STDE at a given time
- as long as it has a finite and well behaved second derivative
- different from STIE - can have only specific eigen functions as solutions
the evolution of a wave can be computed if the wave function is known everywhere at an instant in time
- there are many standard numerical techniques to perform the numerical equations in time

linear superposition

STIE

because STIE is linear in a specific mathematical way
- we can multiply any solution by a constant, and that is still a solution
- this aspect of linearity allows normalization of wave function solutions
a second concept of linearity is linear superposition
- it is the idea that if we have two different solutions of an equation
- then the sum of them is also an equation
linear superposition is the idea that if two different solutions of an equation exist
- then the sum of them is also a solution
- only applicable for degenerate solutions in STIEs
  - degenerate solutions: more than one orthogonal eigenfunction for a given eigenvalue
  - occurs in highly symmetric problems

STDE

STDE is a linear equation
- superposition arises all the time
at a given point in time, any function whose second derivative is well behaved
- to be a solution
because of linearity, the sum of solutions are also solutions
there are a few constraints on our ability to construct linear superpositions of different solutions
- super-positions that will themselves be solutions
- helpful for solving for the possible behaviors in time

linear superposition

STDE is linear in the wavefunction \( \Psi \) \[ -\frac{\hbar^2}{2m} \nabla^2 \Psi(\textbf{r},t) + V(\textbf{r},t) \Psi(\textbf{r},t) = ih\frac{\partial\Psi(\textbf{r},t)}{\partial t} \]
- one reason is that no higher powers of \( \Psi \)
- second is \( \Psi \) appears in every term, there is no additive constant term anywhere
linearity requires two conditions obeyed by STDE
1. if \( \Psi \) is a solution, then so also is \( a\Psi \), where \(a\) is any constant
2. if \( \Psi_a \) and \( \Psi_b \) are solutions, then so also is \( \Psi_a + \Psi_b \)
a consequence of these two conditions is that \[ \Psi_c(\textbf{r},t) = c_a\Psi_a(\textbf{r},t) + c_b\Psi_b(\textbf{r},t) \]
- \( c_a \) and \( c_b \) can be complex constants is a solution
this is the property of linear superposition
- so linear superpositions of solutions of STDE are themselves also solutions
in CM, waves and vibrations are in superpositions
- but in QM, particles can be in superposition
in CM, a particle has a state that is defined by
- its position and
- its momentum
in QM, particles can exist in a superposition of states
- each state can have a different energy, different positions and momenta
for systems changing in time, such super-positions are necessary in QM

time dependence and expansion in eigenstates

if the potential \(V\) is constant in time
- each of the energy eigenstates \( \psi_n(\textbf{r}) \) with eigenstates \(E_n\)
- is separately a solution of STDE
- provided it is multiply by the right complex exponential factor \[ \Psi_n(\textbf{r},t) = \exp{\left( -i\frac{E_nt}{h} \right)} \psi_n(\textbf{r}) \]
so the wavefunction at \( t = 0 \) can be expanded in them \[ \Psi(\textbf{r},0) = \sum_{n}a_n\psi_n(\textbf{r}) \]
- where the \(a_n\) are the expansion coefficients
we also know that a function that starts out as \( \Psi_n(\textbf{r}) \) will evolve in time as \[ \Psi_n(\textbf{r},t) = \exp{ \left( -i\frac{E_n t}{\hbar} \right) } \psi_n(\textbf{r}) \]
- so by linear superposition, the solution at time \(t\) is \[ \begin{align} \Psi(\textbf{r},t) & = \sum_{n}a_n \Psi_n(\textbf{r},t) \\\ & = \sum_n a_n \exp{\left( -i \frac{E_n t}{\hbar} \right)} \psi_n(\textbf{r}) \\
  \end{align} \]
when potential does not vary in time \[ \begin{align} \Psi(\textbf{r},t) & = \sum_{n}a_n\Psi_n(\textbf{r},t) \\
& = \sum_{n}a_n \exp{ \left( -i \frac{E_n t}{\hbar} \right)} \psi(\textbf{r}) \\
\end{align} \]
is the solution of the time-dependent equation
- with the initial condition \[ \begin{align} \Psi(\textbf{r},0) & = \psi(\textbf{r}) \\
  & = \sum_n a_n \psi_n{\textbf(r)} \\
  \end{align} \]
if the wavefunction at time \( t = 0 \) in the energy eigen-states are expanded
- we have solved for the time evolution of the state by adding up the above sum
entire concept of time evolution can be view as a consequence of superposition in QM