# Quantum mechanics

Quantum mechanics is quite a broad term. Perhaps it is best to start approaching it from the division into:
Key experiments that could not work without quantum mechanics: Section "Quantum mechanics experiment".
Mathematics: there are a few models of increasing precision which could all be called "quantum mechanics":
Ciro Santilli feels that the largest technological revolutions since the 1950's have been quantum related, and will continue to be for a while, from deeper understanding of chemistry and materials to quantum computing, understanding and controlling quantum systems is where the most interesting frontier of technology lies.

## Quantum mechanics experiment

Atoms exist and last for a long time, while in classical electromagnetic theory punctual orbiting electrons should emit radiation quickly and fall into the nucleus: physics.stackexchange.com/questions/20003/why-dont-electrons-crash-into-the-nuclei-they-orbit
In other sections:

## Spectral line

A single line in the emission spectrum.
So precise, so discrete, which makes no sense in classical mechanics!
Has been the leading motivation of the development of quantum mechanics, all the way from the:

## NIST Atomic Spectra Database

Let's do a sanity check.
From there we can see for example the 1 to 2 lines:
• 1s to 2p: 121.5673644608 nm
• 1s to 2: 121.56701 nm TODO what does that mean?
• 1s to 2s: 121.5673123130200 TODO what does that mean?
We see that the table is sorted from lower from level first and then by upper level second.
So it is good to see that we are in the same 121nm ballpark as mentioned at hydrogen spectral line.
TODO why I can't see 2s to 2p transitions there to get the fine structure?

## Selection rule

phys.libretexts.org/Courses/University_of_California_Davis/UCD%3A_Physics_9HE_-_Modern_Physics/06%3A_Emission_and_Absorption_of_Photons/6.2%3A_Selection_Rules_and_Transition_Times has some very good mentions:
So it appears that if a hydrogen atom emits a photon, it not only has to transition between two states whose energy difference matches the energy of the photon, but it is restricted in other ways as well, if its mode of radiation is to be dipole. For example, a hydrogen atom in its 3p state must drop to either the n=1 or n=2 energy level, to make the energy available to the photon. The n=2 energy level is 4-fold degenerate, and including the single n=1 state, the atom has five different states to which it can transition. But three of the states in the n=2 energy level have l=1 (the 2p states), so transitioning to these states does not involve a change in the angular momentum quantum number, and the dipole mode is not available.
So what's the big deal? Why doesn't the hydrogen atom just use a quadrupole or higher-order mode for this transition? It can, but the characteristic time for the dipole mode is so much shorter than that for the higher-order modes, that by the time the atom gets around to transitioning through a higher-order mode, it has usually already done so via dipole. All of this is statistical, of course, meaning that in a large collection of hydrogen atoms, many different modes of transitions will occur, but the vast majority of these will be dipole.
It turns out that examining details of these restrictions introduces a couple more. These come about from the conservation of angular momentum. It turns out that photons have an intrinsic angular momentum (spin) magnitude of , which means whenever a photon (emitted or absorbed) causes a transition in a hydrogen atom, the value of l must change (up or down) by exactly 1. This in turn restricts the changes that can occur to the magnetic quantum number: can change by no more than 1 (it can stay the same). We have dubbed these transition restrictions selection rules, which we summarize as:
$$Δl=±1,Δml​=0,±1 (1)$$

## Metastable electron

A fundamental component of three-level lasers.
As mentioned at youtu.be/_JOchLyNO_w?t=581 from Video "How Lasers Work by Scientized (2017)", they stay in that state for a long time compared to non spontaneous emission of metastable states!

## Gross hydrogen emission spectrum

One reasonable and memorable approximation excluding any fine structure is: $$Equation 1. Hydrogen spectral series mnemonic En​=−n213.6eV​ (1)$$ see for example example: hydrogen 1-2 spectral line.

## Hydrogen 1-2 spectral line (121.6 nm, 10.2e V)

Equation "Hydrogen spectral series mnemonic" gives for example from principal quantum number 1 to 2 a difference: $$En​=−13.6eV[221​−121​]=10.2eV (1)$$ which with Planck-Einstein relation gives about 121.6 nm ( Hz), which is a reasonable match with the value of 121.567... from the NIST Atomic Spectra Database.

## Hydrogen spectral series

Kind of a synonym for hydrogen emission spectrum not very clear if fine structure is considered by this term or not.
Formula discovered in 1885, was it the first set to have an empirical formula?

## Fine structure

Split in energy levels due to interaction between electron up or down spin and the electron orbitals.
Numerically explained by the Dirac equation when solving it for the hydrogen atom, and it is one of the main triumphs of the theory.

## Hyperfine structure

Small splits present in all levels due to interaction between the electron spin and the nuclear spin if it is present, i.e. the nucleus has an even number of nucleons.
As the name suggests, this energy split is very small, since the influence of the nucleus spin on the electron spin is relatively small compared to other fine structure.
TODO confirm: does it need quantum electrodynamics or is the Dirac equation enough?
The most important examples:

## Hydrogen line (21 centimeter line)

21 cm is very long and very low energy, because he energy split is very small!
Compare it e.g. with the hydrogen 1-2 spectral line which is 121.6 nm!

## Zeeman effect

Split in the spectral line when a magnetic field is applied.
Non-anomalous: number of splits matches predictions of the Schrödinger equation about the number of possible states with a given angular momentum. TODO does it make numerical predictions?
Anomalous: evidence of spin.
www.pas.rochester.edu/~blackman/ast104/zeeman-split.html contains the hello world that everyone should know: 2p splits into 3 energy levels, so you see 3 spectral lines from 1s to 2p rather than just one.
p splits into 3, d into 5, f into 7 and so on, i.e. one for each possible azimuthal quantum number.
It also mentions that polarization effects become visible from this: each line is polarized in a different way. TODO more details as in an experiment to observe this.

## Double-slit experiment

Amazingly confirms the wave particle duality of quantum mechanics.
The effect is even more remarkable when done with individual particles such individual photons or electrons.
Richard Feynman liked to stress how this experiment can illustrate the core ideas of quantum mechanics. Notably, he night have created the infinitely many slits thought experiment which illustrates the path integral formulation.

## Single particle double slit experiment

This experiment seems to be really hard to do, and so there aren't many super clear demonstration videos with full experimental setup description out there unfortunately.
For single-photon non-double-slit experiments see: single photon production and detection experiments. Those are basically a pre-requisite to this.
photon experiments:
Non-elementary particle:
• 2019-10-08: 25,000 Daltons
• interactive.quantumnano.at/letsgo/ awesome interactive demo that allows you to control many parameters on a lab. Written in Flash unfortunately, in 2015... what a lack of future proofing!

## Are particles bounced by the first wall in the double slit experiment?

It would be amazing to answer this with single particle double slit experiment measurements!

## Quantum Hall effect

Quantum version of the Hall effect.
Gotta understand this because the name sounds cool. Maybe also because it is used to define the fucking ampere in the 2019 redefinition of the SI base units.
At least the experiment description itself is easy to understand. The hard part is the physical theory behind.

## History of quantum mechanics

The discovery of the photon was one of the major initiators of quantum mechanics.
Light was very well known to be a wave through diffraction experiments. So how could it also be a particle???
This was a key development for people to eventually notice that the electron is also a wave.
This process "started" in 1900 with Planck's law which was based on discrete energy packets being exchanged as exposed at On the Theory of the Energy Distribution Law of the Normal Spectrum by Max Planck (1900).
This ideas was reinforced by Einstein's explanation of the photoelectric effect in 1905 in terms of photon.
In the next big development was the Bohr model in 1913, which supposed non-classical physics new quantization rules for the electron which explained the hydrogen emission spectrum. The quantization rule used made use of the Planck constant, and so served an initial link between the emerging quantized nature of light, and that of the electron.
The final phase started in 1923, when Louis de Broglie proposed that in analogy to photons, electrons might also be waves, a statement made more precise through the de Broglie relations.
This event opened the floodgates, and soon matrix mechanics was published in quantum mechanical re-interpretation of kinematic and mechanical relations by Heisenberg (1925), as the first coherent formulation of quantum mechanics.
It was followed by the Schrödinger equation in 1926, which proposed an equivalent partial differential equation formulation to matrix mechanics, a mathematical formulation that was more familiar to physicists than the matrix ideas of Heisenberg.
Inward Bound by Abraham Pais (1988) summarizes his views of the main developments of the subjectit:
• Planck's on the discovery of the quantum theory (1900);
• Einstein's on the light-quantum (1905);
• Bohr's on the hydrogen atom (1913);
• Bose's on what came to be called quantum statistics (1924);
• Heisenberg's on what came to be known as matrix mechanics (1925);
• and Schroedinger's on wave mechanics (1926).

## Introductory Quantum Mechanics by Richard Fitzpatrick (2020)

This LibreTexts book does have some interest!

## MIT 8.06 Quantum Physics III, Spring 2018 by Barton Zwiebach

100 10-20 minute videos properly split by topic, good resource!
Instructor: Barton Zwiebach.
Free material from university courses:

## Applications of Quantum Mechanics by David Tong (2017)

Author: David Tong.
Summary:

## Quantum Mechanics for Engineers by Leon van Dommelen (2011)

Looks very impressive! Last update marked 2011 as of 2020.
Goes up to "A.15 quantum field theory in a Nanoshell", Ciro have to review it to see if there's anything worthwhile in that section.
Personal page says he retired as of 2020: www.eng.fsu.edu/~dommelen/ But hopefully he has more time for these notes!
And he appears to have his own lightweight markup language that transpiles to LaTeX called l2h: www.eng.fsu.edu/~dommelen/l2h/

## Quantum physics by Jim Branson (2003)

For the UCSD Physics 130 course.
Last updated: 2013.
Very good! Goes up to the Dirac equation.

## Mathematical formulation of quantum mechanics

These are the key mathematical ideas to understand!!
There are actually a few formulations out there. By far the dominant one as of 2020 has been the Schrödinger picture, which contrasts notably with the Heisenberg picture.
Another well known one is the de Broglie-Bohm theory, which is deterministic, but non-local.

## Schrödinger picture

To better understand the discussion below, the best thing to do is to read it in parallel with the simplest possible example: Schrödinger picture example: quantum harmonic oscillator.
The state of a quantum system is a vector of unit length in a Hilbert space. TODO why Hilbert Space.
"Making a measurement" for an observable means applying a self-adjoint operator to the state, and after a measurement is done:
• the state collapses to an eigenvector of the self adjoint operator
• the result of the measurement is the eigenvalue of the self adjoint operator
• the probability of a given result happening when the spectrum is discrete is proportional to the modulus of the projection on that eigenvector.
For continuous spectra such as that of the position operator in most systems, e.g. Schrödinger equation for a free one dimensional particle, the projection on each individual eigenvalue is zero, i.e. the probability of one absolutely exact position is zero. To get a non-zero result, measurement has to be done on a continuous range of eigenvectors (e.g. for position: "is the particle present between x=0 and x=1?"), and you have to integrate the probability over the projection on a continuous range of eigenvalues.
In such continuous cases, the probability collapses to an uniform distribution on the range after measurement.
The continuous position operator case is well illustrated at: Video "Visualization of Quantum Physics (Quantum Mechanics) by udiprod (2017)"
Those last two rules are also known as the Born rule.
Self adjoint operators are chosen because they have the following key properties:
• their eigenvalues form an orthonormal basis
• they are diagonalizable
Perhaps the easiest case to understand this for is that of spin, which has only a finite number of eigenvalues. Although it is a shame that fully understanding that requires a relativistic quantum theory such as the Dirac equation.
The next steps are to look at simple 1D bound states such as particle in a box and quantum harmonic oscillator.
The solution to the Schrödinger equation for a free one dimensional particle is a bit harder since the possible energies do not make up a countable set.
This formulation was apparently called more precisely Dirac-von Neumann axioms, but it because so dominant we just call it "the" formulation.
Quantum Field Theory lecture notes by David Tong (2007) mentions that:
if you were to write the wavefunction in quantum field theory, it would be a functional, that is a function of every possible configuration of the field .

## Schrödinger picture example: quantum harmonic oscillator

TODO: use the results from the quantum harmonic oscillator solution to precisely illustrate the discussion at Schrödinger picture with a concrete example.

## Wave function collapse

Similar to quantum jump in the Bohr model, but for the Schrödinger equation.
The idea the the wave function of a small observed system collapses "obviously" cannot be the full physical truth, only a very useful approximation of reality.
Because then are are hard pressed to determine the boundary between what collapses and what doesn't, and there isn't such a boundary, as everything is interacting, including the observer.
The many-worlds interpretation is an elegant explanation for this. Though it does feel a bit sad and superfluous.

## Many-worlds interpretation

One single universal wavefunction, and every possible outcomes happens in some alternate universe. Does feel a bit sad and superfluous, but also does give some sense to perceived "wave function collapse".

## Bra-ket notation

Notation used in quantum mechanics.
Ket is just a vector. Though generally in the context of quantum mechanics, this is an infinite dimensional vector in a Hilbert space like .
Bra is just the dual vector corresponding to a ket, or in other words projection linear operator, i.e. a linear function which can act on a given vector and returns a single complex number. Also known as... dot product.
For example: $$(⟨x∣)∣y⟩ (1)$$ is basically a fancy way of saying: $$x⋅y (2)$$ that is: we are taking the projection of along the direction. Note that in the ordinary dot product notation however, we don't differentiate as clearly what is a vector and what is an operator, while the bra-ket notation makes it clear.
The projection operator is completely specified by the vector that we are projecting it on. This is why the bracket notation makes sense.
It also has the merit of clearly differentiating vectors from operators. E.g. it is not very clear in that is an operator and is a vector, except due to the relative position to the dot. This is especially bad when we start manipulating operators by themselves without vectors.
This notation is widely used in quantum mechanics because calculating the probability of getting a certain outcome for an experiment is calculated by taking the projection of a state on one an eigenvalue basis vector as explained at: Section "Mathematical formulation of quantum mechanics".
Making the projection operator "look like a thing" (the bra) is nice because we can add and multiply them much like we can for vectors (they also form a vector space), e.g.: $$⟨x∣+⟨y∣ (3)$$ just means taking the projection along the direction.
Ciro Santilli thinks that this notation is a bit over-engineered. Notably the bra's are just vectors, which we should just write as usual with ... the bra thing makes it look scarier than it needs to be. And then we should just find a different notation for the projection part.
Maybe Dirac chose it because of the appeal of the women's piece of clothing: bra, in an irresistible call from British humour.
But in any case, alas, we are now stuck with it.

## Dirac-von Neumann axioms

This is basically what became the dominant formulation as of 2020 (and much earlier), and so we just call it the "mathematical formulation of quantum mechanics".

## Phase-space formulation

An "alternative" formulation of quantum mechanics that does not involve operators.

## Non-relativistic quantum mechanics

The first quantum mechanics theories developed.
Their most popular formulation has been the Schrödinger equation.

## Schrödinger equation

Experiments explained:
Experiments not explained: those that the Dirac equation explains like:
To get some intuition on the equation on the consequences of the equation, have a look at:
The easiest to understand case of the equation which you must have in mind initially that of the Schrödinger equation for a free one dimensional particle.
Then, with that in mind, the general form of the Schrödinger equation is: $$Equation 1. Schrodinger equation iℏ∂t∂ψ(x∣,t)​=H^[ψ(x∣,t)] (1)$$ where:
• is the reduced Planck constant
• is the wave function
• is the time
• is a linear operator called the Hamiltonian. It takes as input a function , and returns another function. This plays a role analogous to the Hamiltonian in classical mechanics: determining it determines what the physical system looks like, and how the system evolves in time, because we can just plug it into the equation and solve it. It basically encodes the total energy and forces of the system.
The argument of could be anything, e.g.:
Note however that there is always a single magical time variable. This is needed in particular because there is a time partial derivative in the equation, so there must be a corresponding time variable in the function. This makes the equation explicitly non-relativistic.
The general Schrödinger equation can be broken up into a trivial time-dependent and a time-independent Schrödinger equation by separation of variables. So in practice, all we need to solve is the slightly simpler time-independent Schrödinger equation, and the full equation comes out as a result.

## Time-independent Schrödinger equation

The cool thing about the Time-independent Schrödinger equation is that we can always reduce solving the full Schrödinger equation to solving this slightly simpler time-independent version.
This is actually the approach that we will always take when solving the schrodinger equation, see e.g. quantum harmonic oscillator.
Before reading any further, you must understand heat equation solution with Fourier series, which uses separation of variables.
Once that example is clear, we see that the exact same separation of variables can be done to the Schrödinger equation. If we name the constant of the separation of variables for energy, we get::
• a time-only part that does not depend on space and does not depend on the Hamiltonian at all. The solution for this part is therefore always the same exponentials for any problem, and this part is therefore "boring": $$e−iEt/ℏ (1)$$
• a space-only part that does not depend on time, bud does depend on the hamiltonian:
$$Equation 2. Time-independent Schrodinger equation H^[ψ(x∣)]=Eψx∣ (2)$$
Since this is the only non-trivial part, unlike the time part which is trivial, this spacial part is just called "the time-independent Schrodinger equation".
Note that the here is not the same as the in the time-dependent Schrodinger equation of course, as that psi is the result of the multiplication of the time and space parts. This is a bit of imprecise terminology, but hey, physics.
Because the time part of the equation is always the same and always trivial to solve, all we have to do to actually solve the Schrodinger equation is to solve the time independent one, and then we can construct the full solution trivially.
Once we've solve the time-independent part, call it for each discrete energy as , we proceed exactly as we did in heat equation solution with Fourier series: we make a weighted sum over all possible to match the initial condition, which is analogous to the Fourier series in the case of the heat equation to reach a final full solution: $$Equation 3. Solution of the Schrodinger equation in terms of the time-independent and time dependent parts ∑i=0∞​e−iEt/ℏEi​ψi​(x) (3)$$ and this is a solution by selecting such that at time we match the intial condition: $$∑i=0∞​e−iE0/ℏEi​ψi​(x)=∑i=0∞​Ei​ψi​(x)=initialcondition (4)$$
The fact that this approximation of the initial condition is always possible from is mathematically proven by some version of the spectral theorem bsaed on the fact that The Schrodinger equation Hamiltonian has to be Hermitian and therefore behaves nicely.
It is interesting to note that solving the time-independent Schrodinger equation can also be seen exactly as an eigenvalue equation where:
The only difference from usual matrix eigenvectors is that we are now dealing with an infinite dimensional vector space.
Furthermore:

## Derivation of the Schrodinger equation

Where derivation == "intuitive routes", since a "law of physics" cannot be derived, only observed right or wrong.
Some approaches:

## Solutions of the Schrodinger equation

As always, the best way to get some intuition about an equation is to solve it for some simple cases, so let's give that a try with different fixed potentials.

## Schrödinger equation for a one dimensional particle

We select for the general Equation "Schrodinger equation":
giving the full explicit partial differential equation: $$Equation 1. Schrödinger equation for a one dimensional particle iℏ∂t∂ψ(x,t)​=[−2mℏ2​∂x2∂2​+V(x,t)]ψ(x,t) (1)$$ The corresponding time-independent Schrödinger equation for this equation is: $$Equation 2. time-independent Schrödinger equation for a one dimensional particle [−2mℏ​∂x∂2​+V(x)]ψ=Eψ(x) (2)$$

## Quantum harmonic oscillator

We get the time-independent Schrödinger equation by substituting this into Equation "time-independent Schrödinger equation for a one dimensional particle": $$[−2mℏ​∂x∂2​+x2]ψ=Eψ(x) (1)$$
The first is the stupid "here's a guess" + "hey this family of solutions forms a complete bases"! This is exactly how we solved the problem at Section "Solving partial differential equations with the Fourier series", except that now the complete basis are the Hermite functions.
The second is the much celebrated ladder operator method.

## Quantum LC circuit

A quantum version of the LC circuit!
TODO are there experiments, or just theoretical?

## Hermite polynomials

I.e.: they are both:

## Hermite functions

Not the same as Hermite polynomials.

The operators are a natural guess on the lines of "if p and x were integers".
And then we can prove the ladder properties easily.
The commutator appear in the middle of this analysis.

## Quantum tunnelling

Examples:
• flash memory uses quantum tunneling as the basis for setting and resetting bits
• alpha decay is understood as a quantum tunneling effect in the nucleus

## Schrödinger equation solution for the hydrogen atom

Is the only atom that has a closed form solution, which allows for very good predictions, and gives awesome intuition about the orbitals in general.
It is arguably the most important solution of the Schrodinger equation.
In particular, it predicts:
The explicit solution can be written in terms of spherical harmonics.

## Atomic orbital

In the case of the Schrödinger equation solution for the hydrogen atom, each orbital is one eigenvector of the solution.
Remember from time-independent Schrödinger equation that the final solution is just the weighted sum of the eigenvector decomposition of the initial state, analogously to solving partial differential equations with the Fourier series.
This is the table that you should have in mind to visualize them: en.wikipedia.org/w/index.php?title=Atomic_orbital&oldid=1022865014#Orbitals_table

## Quantum number

Quantum numbers appear directly in the Schrödinger equation solution for the hydrogen atom.
However, it very cool that they are actually discovered before the Schrödinger equation, and are present in the Bohr model (principal quantum number) and the Bohr-Sommerfeld model (azimuthal quantum number and magnetic quantum number) of the atom. This must be because they observed direct effects of those numbers in some experiments. TODO which experiments.
E.g. The Quantum Story by Jim Baggott (2011) page 34 mentions:
As the various lines in the spectrum were identified with different quantum jumps between different orbits, it was soon discovered that not all the possible jumps were appearing. Some lines were missing. For some reason certain jumps were forbidden. An elaborate scheme of ‘selection rules’ was established by Bohr and Sommerfeld to account for those jumps that were allowed and those that were forbidden.
This refers to forbidden mechanism. TODO concrete example, ideally the first one to be noticed. How can you notice this if the energy depends only on the principal quantum number?

## Principal quantum number (n)

Determines energy. This comes out directly from the resolution of the Schrödinger equation solution for the hydrogen atom where we have to set some arbitrary values of energy by separation of variables just like we have to set some arbitrary numbers when solving partial differential equations with the Fourier series. We then just happen to see that only certain integer values are possible to satisfy the equations.

## Azimuthal quantum number (l)

The direction however is not specified by this number.
To determine the quantum angular momentum, we need the magnetic quantum number, which then selects which orbital exactly we are talking about.

## Magnetic quantum number ()

Fixed quantum angular momentum in a given direction.
Can range between .
E.g. consider gallium which is 1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p1:
• the electrons in s-orbitals such as 1s, 2d, and 3d are , and so the only value for is 0
• the electrons in p-orbitals such as 2p, 3p and 4p are , and so the possible values for are -1, 0 and 1
• the electrons in d-orbitals such as 2d are , and so the possible values for are -2, -1, 0 and 1 and 2
The z component of the quantum angular momentum is simply: $$Lz​=ml​ℏ (1)$$ so e.g. again for gallium:
• s-orbitals: necessarily have 0 z angular momentum
• p-orbitals: have either 0, or z angular momentum
Note that this direction is arbitrary, since for a fixed azimuthal quantum number (and therefore fixed total angular momentum), we can only know one direction for sure. is normally used by convention.

## Spectroscopic notation

This notation is cool as it gives the spin quantum number, which is important e.g. when talking about hyperfine structure.
But it is a bit crap that the spin is not given simply as but rather mixes up both the azimuthal quantum number and spin. What is the reason?

Bibliography:

## Solutions of the Schrodinger equation for two electrons

TODO. Can't find it easily. Anyone?
This is closely linked to the Pauli exclusion principle.
What does a particle even mean, right? Especially in quantum field theory, where two electrons are just vibrations of a single electron field.
Another issue is that if we consider magnetism, things only make sense if we add special relativity, since Maxwell's equations require special relativity, so a non approximate solution for this will necessarily require full quantum electrodynamics.
As mentioned at lecture 1 youtube.com/watch?video=H3AFzbrqH68&t=555, relativistic quantum mechanical theories like the Dirac equation and Klein-Gordon equation make no sense for a "single particle": they must imply that particles can pop in out of existence.
Bibliography:

## Orbital approximation

Just ignore the electron electron interactions.

## Schrödinger equation solution for the helium atom

No closed form solution, but good approximation that can be calculated by hand with the Hartree-Fock method, see hartree-Fock method for the helium atom.

## Why do multiple electrons occupy the same orbital if electrons repel each other?

That is, two electrons per atomic orbital, each with a different spin.
As shown at Schrödinger equation solution for the helium atom, they do repel each other, and that affects their measurable energy.
However, this energy is still lower than going up to the next orbital. TODO numbers.
This changes however at higher orbitals, notably as approximately described by the aufbau principle.

## Aufbau principle

Boring rule that says that less energetic atomic orbitals are filled first.
Much more interesting is actually determining that order, which the Madelung energy ordering rule is a reasonable approximation to.