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PERSAMAAN SCRÖDINGER, OPERATOR, DAN PERSAMAAN EIGEN

Dr. Uripto Trisno Santoso

Kimia FMIPA UNLAM2015

WAVE EQUATIONThe wave equation is an important second-order linear partialdifferential equation for the description of waves – as they occur inphysics – such as sound waves, light waves and water waves. Itarises in fields like acoustics, electromagnetics, and fluid dynamics.

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Sinusoidal waves of various frequencies; the bottom waves have higherfrequencies than those above. The horizontal axis represents time.

Spherical waves coming from a point

source.

A pulse traveling through a string with fixedendpoints as modeled by the wave equation.

PERSAMAAN GELOMBANG 1-DIn 1746, d’Alembert discovered the one-dimensional wave equation.

Jean-Baptiste le Rondd'Alembert(16 November 1717 – 29October 1783) was aFrench mathematician,mechanician, physicist,philosopher, and musictheorist.

1-d standing wave as a superposition of two waves traveling in opposite directions

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SCALAR WAVE EQUATION IN ONESPACE DIMENSION

The wave equation in one space dimension can be written like this:

This equation is typically described as having only one space dimension"x", because the only other independent variable is the time "t".Nevertheless, the dependent variable "y" may represent a secondspace dimension, as in the case of a string that is located in the x-yplane.

SCALAR WAVE EQUATION IN TWOSPACE DIMENSION

In two space dimensions, the wave equation is

A solution of the wave equation in two dimensions with a zero-displacement boundary condition along the entire outer edge.

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PERSAMAAN GELOMBANG 3-DIn 1756, Euler discovered the three-dimensional wave equation.

Leonhard Euler(15 April 1707 – 18

September 1783) was a pioneering Swiss

mathematician and physicist.

Cut-away of spherical wavefronts, with awavelength of 10 units, propagating from apoint source.

Erwin Rudolf Josef Alexander Schrödinger (12 August1887 – 4 January 1961), sometimes written as ErwinSchrodinger or Erwin Schroedinger, was a Nobel Prize-winning Austrian physicist who developed a number offundamental results in the field of quantum theory, whichformed the basis of wave mechanics: he formulated thewave equation (stationary and time-dependentSchrödinger equation) and revealed the identity of hisdevelopment of the formalism and matrix mechanics.

• The first publications of Schrödinger about atomic theory and the theory ofspectra began to emerge only from the beginning of the 1920s, after hispersonal acquaintance with Sommerfeld and Wolfgang Pauli and his move toGermany.

• In January 1921, Schrödinger finished his first article on this subject, about theframework of the Bohr-Sommerfeld effect of the interaction of electrons on somefeatures of the spectra of the alkali metals.

• In autumn 1922 he analyzed the electron orbits in an atom from a geometricpoint of view, using methods developed by the mathematician Hermann Weyl.

Erwin Schrödinger

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Schrödinger proposed an original interpretation of the physicalmeaning of the wave function.

• In January 1926, Schrödinger published in Annalen der Physikthe paper "Quantisierung als Eigenwertproblem" [Quantization asan Eigenvalue Problem] on wave mechanics and presented whatis now known as the Schrödinger equation.

• In this paper, he gave a "derivation" of the wave equation fortime-independent systems and showed that it gave the correctenergy eigenvalues for a hydrogen-like atom.

• This paper has been universally celebrated as one of the mostimportant achievements of the twentieth century and created arevolution in quantum mechanics and indeed of all physics andchemistry.

• A second paper was submitted just four weeks later that solved thequantum harmonic oscillator, rigid rotor, and diatomic moleculeproblems and gave a new derivation of the Schrödinger equation.

• A third paper, published in May, showed the equivalence of hisapproach to that of Heisenberg and gave the treatment of the Starkeffect.

• A fourth paper in this series showed how to treat problems in which thesystem changes with time, as in scattering problems. In this paper heintroduced a complex solution to the Wave equation in order to preventthe occurrence of a fourth order differential equation, and this wasarguably the moment when quantum mechanics switched from real tocomplex numbers, never to return.

Schrödinger proposed an original interpretation of the physicalmeaning of the wave function.

These papers were his central achievement and were at once recognized ashaving great significance by the physics community.

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SCHRÖDINGER EQUATION• In quantum mechanics, the Schrödinger equation is a partial

differential equation that describes how the quantum state of aphysical system changes with time. It was formulated in late1925, and published in 1926, by Erwin Schrödinger.

• The most general form is the time-dependent Schrödingerequation, which gives a description of a system evolving withtime:

• where i is the imaginary unit, ħ is the Planck constant divided by 2π (which isknown as the reduced Planck constant), the symbol ∂/∂t indicates a partialderivative with respect to time t, Ψ (the Greek letter Psi) is the wave function ofthe quantum system, and Ĥ is the Hamiltonian operator (which characterizesthe total energy of any given wave function and takes different formsdepending on the situation).

• The most famous example is the non-relativistic Schrödingerequation for a single particle moving in an electric field (but not amagnetic field):

A wave function that satisfies the non-relativistic Schrödinger equation with V = 0. Inother words, this corresponds to a particletraveling freely through empty space.

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TIME-INDEPENDENT EQUATION• The time-independent Schrödinger equation predicts that wave

functions can form standing waves, called stationary states (alsocalled "orbitals", as in atomic orbitals or molecular orbitals).

Ĥ = E

• In words, the equation states:When the Hamiltonian operator acts on a certain wavefunction Ψ, and the result is proportional to the same wavefunction Ψ, then Ψ is a stationary state, and theproportionality constant, E, is the energy of the state Ψ.

• As before, the most famous manifestation is the non-relativistic Schrödinger equation for a single particle movingin an electric field (but not a magnetic field):

• The time-independent Schrödinger equation is the equationdescribing stationary states. (It is only used when theHamiltonian itself is not dependent on time. In general, thewave function still has a time dependency.)

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IMPLICATIONS• The Schrödinger equation, and its solutions, introduced a

breakthrough in thinking about physics.

• Schrödinger's equation was the first of its type, andsolutions led to consequences that were very unusual andunexpected for the time.

1. Total, kinetic, and potential energy

2. Quantization

3. Measurement and uncertainty

4. Quantum tunneling

5. Particles as waves

TOTAL, KINETIC, AND POTENTIAL ENERGY

• The overall form of the equation is not unusual orunexpected as it uses the principle of the conservation ofenergy.

• The terms of the nonrelativistic Schrödinger equation can beinterpreted as total energy of the system, equal to thesystem kinetic energy plus the system potential energy.

• In this respect, it is just the same as in classical physics.

P.R.Tunjukkan konsistensi persamaan Scrödinger dengan konsepkekelan energy menurut mekanika klasik.

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QUANTIZATION• The Schrödinger equation predicts that if certain properties of a

system are measured, the result may be quantized, meaning thatonly specific discrete values can occur. One example is energyquantization: the energy of an electron in an atom is always oneof the quantized energy levels, a fact discovered via atomicspectroscopy. (Energy quantization is discussed below.)

• Another example is quantization of angular momentum. This wasan assumption in the earlier Bohr model of the atom, but it is aprediction of the Schrödinger equation.

• Another result of the Schrödinger equation is that not everymeasurement gives a quantized result in quantum mechanics.For example, position, momentum, time, and (in some situations)energy can have any value across a continuous range.

MEASUREMENT AND UNCERTAINTY• In classical mechanics, a particle has, at every moment, an exact position

and an exact momentum. These values change deterministically as theparticle moves according to Newton's laws. In quantum mechanics,particles do not have exactly determined properties, and when they aremeasured, the result is randomly drawn from a probability distribution.The Schrödinger equation predicts what the probability distributions are,but fundamentally cannot predict the exact result of each measurement.

• The Heisenberg uncertainty principle is the statement of the inherentmeasurement uncertainty in quantum mechanics. It states that the moreprecisely a particle's position is known, the less precisely its momentum isknown, and vice versa.

• The Schrödinger equation describes the (deterministic) evolution of thewave function of a particle. However, even if the wave function is knownexactly, the result of a specific measurement on the wave function isuncertain.

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QUANTUM TUNNELING• In classical physics, when a ball is rolled slowly up a

large hill, it will come to a stop and roll back, because itdoesn't have enough energy to get over the top of thehill to the other side.

• However, the Schrödinger equation predicts that thereis a small probability that the ball will get to the otherside of the hill, even if it has too little energy to reachthe top. This is called quantum tunneling.

• It is related to the uncertainty principle: Although theball seems to be on one side of the hill, its position isuncertain so there is a chance of finding it on the otherside.

Quantum tunneling through a barrier. A particle coming from the left does not have enough energy to climb the barrier. However, it can sometimes "tunnel" to the other side.

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PARTICLES AS WAVES• The nonrelativistic Schrödinger equation is a type of partial

differential equation called a wave equation. Therefore it isoften said particles can exhibit behavior usually attributed towaves. In most modern interpretations this description isreversed – the quantum state, i.e. wave, is the only genuinephysical reality, and under the appropriate conditions it canshow features of particle-like behavior.

• Two-slit diffraction is a famous example of the strangebehaviors that waves regularly display, that are not intuitivelyassociated with particles. The overlapping waves from thetwo slits cancel each other out in some locations, andreinforce each other in other locations, causing a complexpattern to emerge. Intuitively, one would not expect thispattern from firing a single particle at the slits, because theparticle should pass through one slit or the other, not acomplex overlap of both.

A double slit experiment showing theaccumulation of electrons on a screenas time passes.• a single particle fired through a

double-slit does show this samepattern (figure on right). Note: Theexperiment must be repeated manytimes for the complex pattern toemerge.

• The appearance of the patternproves that each electron passesthrough both slits simultaneously.

• Although this is counterintuitive, theprediction is correct; in particular,electron diffraction and neutrondiffraction are well understood andwidely used in science andengineering.

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PROPERTIES1. Linearity

In the development above, the Schrödinger equation was made to be linearfor generality, though this has other implications. If two wave functions ψ1 andψ2 are solutions, then so is any linear combination of the two:

where a and b are any complex numbers (the sum can be extended for anynumber of wavefunctions).

2. Real energy eigenstates

For the time-independent equation, an additional feature of linearity follows: iftwo wave functions ψ1 and ψ2 are solutions to the time-independent equationwith the same energy E, then so is any linear combination:

Two different solutions with the same energy are called degenerate

PROPERTIES3. Space and time derivatives

The Schrödinger equation is first order in time and second in space, whichdescribes the time evolution of a quantum state (meaning it determines thefuture amplitude from the present).Explicitly for one particle in 3-dimensional Cartesian coordinates – the equationis

The first time partial derivative implies the initial value (at t = 0) of thewavefunction

is an arbitrary constant. Likewise – the second order derivatives with respectto space implies the wavefunction and its first order spatial derivatives

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PROPERTIES4. Local conservation of probability

The Schrödinger equation is consistent with probability conservation.Multiplying the Schrödinger equation on the right by the complex conjugatewavefunction, and multiplying the wavefunction to the left of the complexconjugate of the Schrödinger equation, and subtracting, gives the continuityequation for probability:

Where:

is the probability density (probability per unit volume, * denotes complexconjugate), and

is the probability current (flow per unit area).

Hence predictions from the Schrödinger equation do not violateprobability conservation.

Using the Schrödinger equation to develop the de Broglie relation

• The Schrödinger equation can be seen to be plausible by notingthat it implies the de Broglie relation for a freely moving particle ina region where its potential energy V is constant.

After writing V(x) V, we can rearrange

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• We now recognize that cos kx is a wave of wavelength λ = 2π/k, ascan be seen by comparing cos kx with the standard form of aharmonic wave, cos(2πx/λ).

• The quantity E−V is equal to the kinetic energy of the particle, Ek,so k = (2mEk/ħ2)1/2, which implies that Ek = k2ħ2/2m.

• Because Ek = p2/2m, it follows that p = kħ.

• Therefore, the linear momentum is related to the wavelength of thewavefunction by

which is the de Broglie relation

• The Schrödinger equation itself also implies somemathematical restrictions on the type of functions thatwill occur. Because it is a second-order differentialequation, the second derivative of ψ must be well-defined if the equation is to be applicable everywhere.We can take the second derivative of a function only if itis continuous and if its first derivative, its slope, iscontinuous.

• At this stage we see that must be:1. continuous2. have a continuous slope3. be single-valued4. be square-integrable

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The wavefunction must satisfy stringent conditions for it to beacceptable. (a) Unacceptable because it is not continuous; (b)unacceptable because its slope is discontinuous; (c) unacceptablebecause it is not single-valued; (d) unacceptable because it isinfinite over a finite region.

THE BORN INTERPRETATION OF THE WAVEFUNCTION

Key points:1. According to the Born interpretation, the probability density

is proportional to the square of the wavefunction.2. A wavefunction is normalized if the integral of its square is

equal to 1.3. The quantization of energy stems from the constraints that

an acceptable wavefunction must satisfy.

For a one-dimensional system:If the wavefunction of a particle has the value ψ at some point x,then the probability of finding the particle between x and x + dx isproportional to |ψ |2dx.

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• If the wavefunction of a particle has the value ψ at some point r,then the probability of finding the particle in an infinitesimalvolume dτ = dxdydz at that point is proportional to |ψ |2dτ.

Normalization

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Therefore, for this integral to equal 1, we must set

and the normalized wavefunction is

CONTOH 1

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OPERATOR• The actions have the names integrate, form the inverse, multiply,

and add, and they are all called operators.

• To every measurable quantity (observable), such as energy,momentum, or position, there is a corresponding operator inquantum mechanics. Quantum mechanical operators usuallyinvolve differentiation with respect to a variable such as x ormultiplication by x or a function of the energy such as V(x).Operators are denoted by a caret: Ô.

• an operator Ô has a set of eigenfunctions and eigenvalues.

The operator acting on these special wave functions returns thewave function multiplied by a number. These special functions arecalled the eigenfunctions of the operator and the an are called theeigenvalues.

Contoh 2

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ORTHOGONALITY• The Eigenfunctions of a Quantum Mechanical Operator Are

Orthogonal.

• We are familiar with the concept of orthogonal vectors. Forexample, orthogonality in three-dimensional Cartesiancoordinate space is defined by

• in which the scalar product between the unit vectors along the x,y, and z axes is zero.

• The analogous expression that defines orthogonalitybetween the eigenfunctions i(x) and j(x) of a quantummechanical operator is

CONTOH 3

• The functions are shown in the following graphs.• The vertical axes have been offset to avoid overlap and

the horizontal line indicates the zero for each plot.• Because the functions are periodic, we can draw

conclusions about their behavior in a very large intervalthat is an integral multiple of by considering theirbehavior in any interval that is an integral multiple of theperiod.

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