[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

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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 $$E=\hslash \omega$$
  • $\omega$: corresponding angular constant
  • $\hslash$: planck’s constant for wavelength relationship

time dependent equation

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

schrodinger’s time-dependent equation

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

sign convention

• schrodinger chose a sign for the right hand side
  • which means that a wave with a spatial part $\propto \exp \left(ikz\right)$
  • is definitely going in the positive $z$ direction
• including its time dependence, that wave would be of the form
  • $\exp \left[i\right(kz-\frac{Et}{\hslash }\left)\right]$
  • 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 $\psi \left(\text{r}\right)$ for the eigen energy $E$ $$-\frac{{\hslash }^2}{2m}{\nabla }^2\psi \left(\text{r}\right)+V\left(\text{r}\right)\psi \left(\text{r}\right)=E\psi \left(\text{r}\right)$$

• however, this solution $\psi \left(\text{r}\right)$ is not a solution of the time-dependent equation $$-\frac{{\hslash }^2}{2m}{\nabla }^2\Psi (\text{r},t)+V(\text{r},t)\Psi (\text{r},t)=ih\frac{\partial \Psi (\text{r},t)}{\partial t}$$

• $\psi \left(\text{r}\right)$ is here for $\Psi (\text{r},t)$ does not work because
  • $\psi \left(\text{r}\right)$ 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 $E\psi \left(\text{r}\right)$

• instead of using solution $\psi \left(\text{r}\right)$, use $$\begin{array}{cc}\Psi (\text{r},t)& =\psi \left(\text{r}\right)\exp (-i\frac{Et}{\hslash })\\ -\frac{h^2}{2m}{\nabla }^2\Psi (\text{r},t)+V\left(\text{r}\right)\Psi (\text{r},t)& =-\frac{{\hslash }^2}{2m}{\nabla }^2\psi \left(\text{r}\right)\exp (-i\frac{Et}{\hslash })+V\left(\text{r}\right)\psi \left(\text{r}\right)\exp (-i\frac{Et}{\hslash })\\ & =[-\frac{{\hslash }^2}{2m}{\nabla }^2\psi (\text{r})+V(\text{r}\left)\psi \right(\text{r}\left)\right]\exp (-i\frac{Et}{\hslash })\\ & =E\psi \left(\text{r}\right)\exp (-i\frac{Et}{\hslash })\\ & =E\Psi (\text{r},t)\end{array}$$

• so the expression $$\Psi (\text{r},t)=\psi \left(\text{r}\right)\exp (-i\frac{Et}{\hslash })$$
  • solves the time-dependent schrodinger equation
• similarly, knowing that $\psi \left(\text{r}\right)$
  • solves the time-INdependent equation with energy $E$
  • substituting $\Psi (\text{r},t)=\psi \left(\text{r}\right)\exp (-i\frac{Et}{h})$
    • in the ime-dependent equation gives $$\begin{array}{cc}-\frac{{\hslash }^2}{2m}{\nabla }^2\Psi (\text{r},t)+V\left(\text{r}\right)\Psi (\text{r},t)& =i\hslash \frac{\partial \Psi (\text{r},t)}{\partial t}\\ & =i\hslash \frac{\partial }{\partial t}\left[\psi \right(\text{r}\left)\exp (-i\frac{Et}{\hslash })\right]\\ & =i\hslash \psi \left(\text{r}\right)\frac{\partial }{\partial t}\left[\exp (-i\frac{Et}{\hslash })\right]\\ & =i\hslash \psi \left(\text{r}\right)[-i\frac{E}{\hslash }]\exp (-i\frac{Et}{\hslash })\\ & =E\Psi (\text{r},t)\end{array}$$
  • so $\Psi (\text{r},t)=\psi \left(\text{r}\right)\exp (-i\frac{Et}{\hslash })$
    • solves the time-dependent schrodinger equation
    • as long as we always multiply it by a factor $\exp (-i\frac{Et}{h})$

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time-dependent from time-INdependent solutions

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

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oscillations and time-independence

• if the following is a proposed solution to a time-independent problem $$\Psi (\text{r},t)=\psi \left(\text{r}\right)\exp (-i\frac{Et}{\hslash })$$
  • 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 $$\begin{array}{cc}\left|\Psi \right(\text{r},t){|}^2& =\left[\exp \left(i\frac{Et}{\hslash }\right){\psi }^{\ast }\right(\text{r}\left)\right]\times \left[\exp (-i\frac{Et}{\hslash })\psi \right(\text{r}\left)\right]\\ & =\left|\psi \right(\text{r}){|}^2\end{array}$$

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

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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 $$f\propto \exp \left[i\right(kz-\omega t\left)\right]$$
  • 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 $$-\frac{h^2}{2m}{\nabla }^2\Psi (\text{r},t)+V(\text{r},t)\Psi (\text{r},t)=i\hslash \frac{\partial \Psi (\text{r},t)}{\partial t}$$
  • $\Psi$ is required to be a complex entity
• for example $V=0$
  • though $\exp \left[i\right(kz-\frac{Et}{h}\left)\right]$ is a solution
  • $\sin (kz-\frac{Et}{\hslash })$ 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

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time evolution form SC

• consider STDE $$-\frac{{\hslash }^2}{2m}{\nabla }^2\Psi (\text{r},t)+V(\text{r},t)\Psi (\text{r},t)=ih\frac{\partial \Psi (\text{r},t)}{\partial t}$$
• if we knew the wavefunction $\Psi (\text{r},t)$ at every point in space at some time $t_0$
  • the LHS of the STDE can be computed at that time for all $\text{r}$
  • so we would know $\frac{\partial \Psi (\text{r},t)}{\partial t}$ for all $\text{r}$
  • so we could integrate the equation to deduce $\Psi (\text{r},t)$ at all future time
• explicitly, knowing $\left(\frac{\partial \Psi (\text{r},t)}{\partial t}\right)$
  • calculate a future wave; for instance, a small instance $\delta t$ later in time $$\Psi (\text{r},t_0+\delta t)\cong \Psi (\text{r},t_0)+\frac{\partial \Psi }{\partial t}{|}_{\text{r},t_0}\delta t$$
  • 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

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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 $\Psi$ $$-\frac{{\hslash }^2}{2m}{\nabla }^2\Psi (\text{r},t)+V(\text{r},t)\Psi (\text{r},t)=ih\frac{\partial \Psi (\text{r},t)}{\partial t}$$
  • 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 $${\Psi }_c(\text{r},t)=c_a{\Psi }_a(\text{r},t)+c_b{\Psi }_b(\text{r},t)$$
  • $c_a$ 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 ${\psi }_n\left(\text{r}\right)$ with eigenstates $E_n$
  • is separately a solution of STDE
  • provided it is multiply by the right complex exponential factor $${\Psi }_n(\text{r},t)=\exp (-i\frac{E_{n}t}{h}){\psi }_n\left(\text{r}\right)$$
• so the wavefunction at $t=0$ can be expanded in them $$\Psi (\text{r},0)=\sum _n{a}_n{\psi }_n\left(\text{r}\right)$$
  • where the $a_n$ are the expansion coefficients
• we also know that a function that starts out as ${\Psi }_n\left(\text{r}\right)$ will evolve in time as $${\Psi }_n(\text{r},t)=\exp (-i\frac{E_{n}t}{\hslash }){\psi }_n\left(\text{r}\right)$$
  • so by linear superposition, the solution at time $t$ is $$\begin{array}{cc}\Psi (\text{r},t)& =\sum _n{a}_n{\Psi }_n(\text{r},t)\\ & =\sum _n{a}_n\exp (-i\frac{E_{n}t}{\hslash }){\psi }_n\left(\text{r}\right)\end{array}$$
• when potential does not vary in time $$\begin{array}{cc}\Psi (\text{r},t)& =\sum _n{a}_n{\Psi }_n(\text{r},t)\\ & =\sum _n{a}_n\exp (-i\frac{E_{n}t}{\hslash })\psi \left(\text{r}\right)\end{array}$$
• is the solution of the time-dependent equation
  • with the initial condition $$\begin{array}{cc}\Psi (\text{r},0)& =\psi \left(\text{r}\right)\\ & =\sum _n{a}_n{\psi }_n\text{(}r)\end{array}$$
• 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