Wave function in quantum mechanics pdf

Things on a very small scale behave like nothing that you have any direct experience about. They do not behave like waves, they do not behave like particles, they do not behave like clouds, or billiard balls, or weights on springs, or like anything that you have ever seen.

Midterm Exam Monday, Feb. It will cover lectures and some aspects of lectures Practice exams: Old exams are linked from the course web page. Review Sunday, Feb. Next week: Homework 4 covers material in lecture 10 — due on Thur.

We strongly encourage you to look at the homework before the midterm! Discussion: Covers material in lectures There will be a quiz.

Lab: Go to Loomis a computer room. Usually the conditions are specified by giving a potential energy U x,y,z in which the particle is located. U x Classically, a particle in For simplicity, the lowest energy state consider a would sit right at the 1-dimensional potential energy bottom of the well.

In QM function, U x. At some random time later, what is the probability of finding it near position x? There are two important forms for the SEQ. Time does not appear in the equation. Therefore, y x,y,z is a standing wave, because the probability density, y x 2, Notation: is not a function of time. Distinguish Y x,y,z,t from y x,y,z. Can we understand the KE term? Consider a particle with a definite momentum.

This is sometimes useful when analyzing a problem. The corresponding probability distributions y x 2 of these states are: y 2 y 2 y 2. Which of these wavefunctions represents the particle with the lowest kinetic energy?

Which corresponds to the highest kinetic energy? Since b clearly has the least curvature, that particle has lowest KE. In this case you can also 2. Probability distribution Difference between classical and quantum cases Classical Quantum U x particle with same energy U x lowest energy state state as in qunatum case. For positive C i. Most of the wave functions in P will be sinusoidal or exponential.

However, we can constrain 0 y more than this. L www. Web site newt. However, in many problems, including the 1D box, U x has different functional forms in different regions.

In our box problem, there are three regions:. We must make sure that y x satisfies the constraints e. They appear in many problems. The waves have exactly the same form as standing waves on a string, sound waves in a pipe, etc. On a string the wave is a displacement y x and the square is the intensity, etc.

The discrete set of allowed wavelengths results in a discrete set of tones that the string can produce. You see how to handle many electrons in atoms, particles interacting with other particles, and particles that scatter off other particles. In Part V, you see how quantum physics can handle the situation. This part is made up of fast-paced lists of ten items each. This icon marks something to remember, such as a law of physics or a particu- larly juicy equation.

This icon means that what follows is technical, insider stuff. This icon helps you avoid mathematical or conceptual slip-ups. You can jump in anywhere you like. And if your upcoming vacation to Geneva, Switzerland, includes a side trip to your new favorite particle accelerator — the Large Hadron Collider — you can flip to Chapter 12 and read up on scattering theory. T his part is designed to give you an introduction to the ways of quantum physics.

You see the issues that gave rise to quantum physics and the kinds of solutions it provides. I also introduce you to the kind of math that quantum physics requires, including the notion of state vectors. Chapter 1. A ccording to classical physics, particles are particles and waves are waves, and never the twain shall mix. But the reality is different — particles turn out to exhibit wave-like proper- ties, and waves exhibit particle-like properties as well.

The idea that waves like light can act as particles like electrons and vice versa was the major revelation that ushered in quantum physics as such an important part of the world of physics. This chapter takes a look at the challenges facing classical physics around the turn of the 20th century — and how quantum physics gradually came to the rescue.

Up to that point, the classical way of looking at physics was thought to explain just about everything. That made the theoretical physicists mad, and they got on the job.

Essential Quantum Physics. Taking a look at how classical physics col- lapsed gives you an introduction to quantum physics that shows why people needed it. Being Discrete: The Trouble with Black-Body Radiation One of the major ideas of quantum physics is, well, quantization — measuring quantities in discrete, not continuous, units. The idea of quantized energies arose with one of the earliest challenges to classical physics: the problem of black-body radiation.

When you heat an object, it begins to glow. The reason it glows is that as you heat it, the electrons on the surface of the material are agitated thermally, and electrons being accelerated and decelerated radiate light.

Physics in the late 19th and early 20th centuries was concerned with the spectrum of light being emitted by black bodies. A black body is a piece of material that radiates corresponding to its temperature — but it also absorbs and reflects light from its surroundings. To make matters easier, physics pos- tulated a black body that reflected nothing and absorbed all the light falling on it hence the term black body, because the object would appear perfectly black as it absorbed all light falling on it.

When you heat a black body, it would radiate, emitting light. But the physicists were clever about this, and they came up with the hollow cavity you see in Figure , with a hole in it. When you shine light on the hole, all that light would go inside, where it would be reflected again and again — until it got absorbed a negligible amount of light would escape through the hole.

And when you heated the hollow cavity, the hole would begin to glow. So there you have it — a pretty good approximation of a black body. Chapter 1: Discoveries and Essential Quantum Physics You can see the spectrum of a black body and attempts to model that spec- trum in Figure , for two different temperatures, T1 and T2.

The problem was that nobody was able to come up with a theoretical explanation for the spectrum of light generated by the black body. Everything classical physics could come up with went wrong. Figure Black-body radiation spectrum. Using classical thermodynamics, he came up with this formula:. Second attempt: Rayleigh-Jeans Law Next up in the attempt to explain the black-body spectrum was the Rayleigh- Jeans Law, introduced around This law predicted that the spectrum of a black body was.

This was called the ultraviolet catastrophe because the best predictions available diverged at high frequencies corresponding to ultraviolet light. It was time for quantum physics to take over. With this theory, crazy as it sounded in the early s, Planck converted the continuous integrals used by Rayleigh-Jeans to discrete sums over an infinite number of terms.

Making that simple change gave Planck the following equa- tion for the spectrum of black-body radiation:. This equation got it right — it exactly describes the black-body spectrum, both at low and high and medium, for that matter frequencies.

This idea was quite new. Saying that the energy of all oscillators was quantized was the birth of quantum physics. Oscillators can oscillate only at discrete energies?

Where did that come from? In any case, the revolution was on — and there was no stopping it. Light, it turns out, exhibits properties of both waves and particles.

This section shows you some of the evidence. Solving the photoelectric effect The photoelectric effect was one of many experimental results that made up a crisis for classical physics around the turn of the 20th century. When you shine light onto metal, as Figure shows, you get emitted electrons. According to clas- sical physics, light is just a wave, and it can exchange any amount of energy with the metal. When you beam light on a piece of metal, the electrons in the metal should absorb the light and slowly get up enough energy to be emit- ted from the metal.

The idea was that if you were to shine more light onto the metal, the electrons should be emitted with a higher kinetic energy. In fact, no matter how weak the intensity of the incident light and researchers tried experiments with such weak light that it should have taken hours to get any electrons emitted , electrons were emit- ted. Light Electrons. Figure The photo- electric effect. Experiments with the photoelectric effect showed that the kinetic energy, K, of the emitted electrons depended only on the frequency — not the intensity — of the incident light, as you can see in Figure Figure Kinetic energy of emitted electrons versus frequency of the inci- dent light.

The results were hard to explain classically, so enter Einstein. This was the beginning of his heyday, around Light, he sug- gested, acted like particles as well as waves. So in this scheme, when light hits a metal surface, photons hit the free elec- trons, and an electron completely absorbs each photon. That is,. Solving for K gives you the following:. In other words, light is quantized. That was also quite an unexpected piece of work by Einstein, although it was based on the earlier work of Planck.

Light quantized? Light coming in discrete energy packets? What next? Scattering light off electrons: The Compton effect To a world that still had trouble comprehending light as particles see the preceding section , Arthur Compton supplied the final blow with the Compton effect. His experiment involved scattering photons off electrons, as Figure shows. Figure Light incident on an electron Photon Electron at rest at rest. After that happens, the light is scattered, as you see in Figure Arthur Compton could explain the results of his experiment only by making the assumption that he was actually dealing with two particles — a photon and an electron.

That is, he treated light as a discrete particle, not a wave. And he made the assumption that the photon and the electron collided elastically — that is, that both total energy and momentum were conserved. And experi- ment confirms this relation — both equations. Note that to derive the wavelength shift, Compton had to make the assump- tion that here, light was acting as a particle, not as a wave. That is, the par- ticle nature of light was the aspect of the light that was predominant. Proof positron?

Dirac and pair production In , the physicist Paul Dirac posited the existence of a positively charged anti-electron, the positron. It was a bold prediction — an anti-particle of the electron? But just four years later, physicists actually saw the positron. In those days, physicists relied on cosmic rays — those particles and high- powered photons called gamma rays that strike the Earth from outer space — as their source of particles.

They used cloud-chambers, which were filled with vapor from dry ice, to see the trails such particles left. They put their chambers into magnetic fields to be able to measure the momentum of the particles as they curved in those fields. In , a physicist noticed a surprising event. A pair of particles, oppositely charged which could be determined from the way they curved in the mag- netic field appeared from apparently nowhere.

No particle trail led to the origin of the two particles that appeared. That was pair-production — the con- version of a high-powered photon into an electron and positron, which can happen when the photon passes near a heavy atomic nucleus.

So experimentally, physicists had now seen a photon turning into a pair of particles. As if everyone needed more evidence of the particle nature of light. Later on, researchers also saw pair annihilation: the conversion of an electron and positron into pure light. At this point, there was an abundance of evidence of the particle-like aspects of light.

A Dual Identity: Looking at Particles as Waves In , the physicist Louis de Broglie suggested that not only did waves exhibit particle-like aspects but the reverse was also true — all material par- ticles should display wave-like properties.

De Broglie said that the same relation should hold for all material particles. De Broglie presented these apparently surprising suggestions in his Ph.

Researchers put these suggestions to the test by sending a beam through a dual-slit apparatus to see whether the electron beam would act like it was made up of particles or waves. In Figure , you can see the setup and the results. Figure An electron beam going through two slits.

In Figure a, you can see a beam of electrons passing through a single slit and the resulting pattern on a screen. In Figure b, the electrons are pass- ing through a second slit. Experiment bore out the relation that , and de Broglie was a success. But if you have an electron, which is it — a wave or a particle?

The act of measurement is what brings out the wave or particle properties. You see more about this idea throughout the book. Quantum mechanics lives with an uncertain picture quite happily. And knowing that inspired Werner Heisenberg, in , to come up with his celebrated uncertainty principle.

You can completely describe objects in classical physics by their momentum and position, both of which you can measure exactly. In other words, classi- cal physics is completely deterministic. On the atomic level, however, quantum physics paints a different picture. That is to say, the more accurately you know the position of a particle, the less accurately you know the momentum, and vice versa.

This relation holds for all three dimensions:. Quantum physics, unlike classical physics, is completely undeterministic. You can never know the precise position and momentum of a particle at any one time.

You can give only probabilities for these linked measurements. The wave function describes the de Broglie wave of a particle, giving its amplitude as a function of position and time.

This book is largely a study of the wave function — the wave functions of free particles, the wave functions of particles trapped inside potentials, of identi- cal particles hitting each other, of particles in harmonic oscillation, of light scattering from particles, and more. Using this kind of physics, you can pre- dict the behavior of all kinds of physical systems. Chapter 2. Sometimes, you get to do things that are a little more mundane, like turn lights off and on, perform a bit of calculus, or play with dice.

In quantum physics, absolute measurements are replaced by probabilities, so you may use dice to calculate the probabilities that various numbers will come up. You can then assemble those values into a vector single-column matrix in Hilbert space a type of infinitely dimensional vector space with some properties that are especially valuable in quantum physics.

This chapter introduces how you deal with probabilities in quantum phys- ics, starting by viewing the various possible states a particle can occupy as a vector — a vector of probability states. From there, I help you familiarize yourself with some mathematical notations common in quantum physics, including bras, kets, matrices, and wave functions.

Along the way, you also get to work with some important operators. You come up with a list indicating the relative probability of rolling a 2, 3, 4, and so on, all the way up to You have a vector of the probabilities that the dice will occupy various states.

To find the actual probability that a particle will be in a certain state, you add wave functions — which are going to be represented by these vectors — and then square them see Chapter 1 for info on why. Making Life Easier with Dirac Notation When you have a state vector that gives the probability amplitude that a pair of dice will be in their various possible states, you basically have a vector in dice space — all the possible states that a pair of dice can take, which is an dimensional space.

See the preceding section for more on state vectors. But in most quantum physics problems, the vectors can be infinitely large — for example, a moving particle can be in an infinite number of states. So in the dice example, you can write the state vector as a ket this way:. Here, the components of the state vector are represented by numbers in dimensional dice space.

A complex conjugate flips the sign connecting the real and imaginary parts of a complex number. This is just matrix multiplication, and the result is the same as taking the sum of the squares of the elements:.

Therefore, in general, the product of the bra and ket equals You may want to, for example, represent your states in a three-dimensional momentum space, with three axes in Hilbert space, px , py , and pz. That is, you can perform your calculations in purely sym- bolic terms, without being tied to a basis. Understanding some relationships using kets Ket notation makes the math easier than it is in matrix form because you can take advantage of a few mathematical relationships.

This turns out the be the analog of the vector inequality:. So why is the Schwarz inequality so useful? It turns out that you can derive the Heisenberg uncertainty principle from it see Chapter 1 for more on this principle. Other ket relationships can also simplify your calculations. And the biggest such prediction is the expectation value. The expectation value of an operator is the aver- age value that you would measure if you performed the measurement many times. In this dice example, the expectation value is a sum of terms, and each term is a value that can be displayed by the dice, multiplied by the probability that that value will appear.

Spelling that out in terms of components gives you the following:. So the expectation value of a roll of the dice is 7. Looking at linear operators An operator A is said to be linear if it meets the following condition:. In order for us to see this we shall need to know just a little more about what happens when we take the products of bras and kets. Now that you know how the product of a bra with a sum of two kets goes, you can say,.

To find the Hermitian adjoint, follow these steps:. Replace complex constants with their complex conjugates. The Hermitian adjoint of a complex number is the complex conjugate of that number:.

Replace kets with their corresponding bras, and replace bras with their corresponding kets. You have to exchange the bras and kets when finding the Hermitian adjoint of an operator, so finding the Hermitian adjoint of an operator is not just the same as mathematically finding its complex conjugate.

Replace operators with their Hermitian adjoints. In quantum mechanics, operators that are equal to their Hermitian adjoints are called Hermitian operators. In other words, an operator is Hermitian if.

Hermitian operators appear throughout the book, and they have spe- cial properties. Write your final equation. First, write the adjoint:. A and B here are Hermitian operators. When you take the Hermitian adjoint of an expression and get the same thing back with a negative sign in front of it, the expression is called anti-Hermitian, so the commutator of two Hermitian operators is anti-Hermitian.

And by the way, the expectation value of an anti- Hermitian operator is guaranteed to be completely imaginary. How can you put it to work? You can come up with the Heisenberg uncertainty relation starting virtually from scratch. This kind of calculation shows how much easier it is to use the basis-less bra and ket notation than the full matrix ver- sion of state vectors. Similarly, the uncer- tainty in a measurement using Hermitian operator B is. So you can rewrite the Schwarz inequality like this:.

Okay, where has this gotten you? The commutator of two Hermitian operators, [A, B], is anti-Hermitian. And because the second term on the right is positive or zero, you can say that the following is true:.

You want to reproduce the Heisenberg uncertainty relation here, which looks like this:. Well, well, well. Is that the Heisenberg uncertainty relation? Well, take a look. In quan- tum mechanics, the momentum operator looks like this:. Hot dog! That is the Heisenberg uncertainty relation. Because R is a diagonal matrix, finding the eigenvectors is easy. You can take unit vectors in the eleven different directions as the eigenvectors. And the eigenvalues? Because the eigenvectors are just unit vectors in all 11 dimensions, the eigenvalues are the numbers on the diagonal of the R matrix: 2, 3, 4, and so on, up to Understanding how they work The eigenvectors of a Hermitian operator define a complete set of orthonormal vectors — that is, a complete basis for the state space.

You can see why the term eigen is applied to eigenvectors — they form a natural basis for the operator. If two or more of the eigenvalues are the same, that eigenvalue is said to be degenerate. So for example, if three eigenvalues are equal to 6, then the eigen- value 6 is threefold degenerate. Finding eigenvectors and eigenvalues So given an operator in matrix form, how do you find its eigenvectors and eigenvalues?

This is the equation you want to solve:. Try to find the eigenvalues and eigenvectors of the follow- ing matrix:. Finding eigenvectors How about finding the eigenvectors? To find the eigenvector corresponding to a1 see the preceding section , substitute a1 — the first eigenvalue, —2 — into the matrix in the form A — aI:. And that means that, up to an arbitrary constant, the eigenvector corre- sponding to a1 is the following:.

How about the eigenvector corresponding to a2? Plugging a2, —3, into the matrix in A —aI form, you get the following:. And that means that, up to an arbitrary con- stant, the eigenvector corresponding to a2 is. Preparing for the Inversion: Simplifying with Unitary Operators Applying the inverse of an operator undoes the work the operator did:.

To find the adjoint of an operator, A, you find the transpose by interchanging the rows and columns, AT. This gives you the following equation:. One of the central problems of quantum mechanics is to calculate the energy levels of a system. The energy operator is called the Hamilitonian, H, and finding the energy levels of a system breaks down to finding the eigenvalues of the problem:. The allowable energy levels of the physical system are the eigenvalues E, which satisfy this equation.

These can be found by solving the characteristic polynomial, which derives from setting the determinant of the above matrix to zero, like so. But what if the number of energy states is infinite?

In that case, you can no longer use a discrete basis for your operators and bras and kets — you use a continuous basis. In the continuous basis, summations become integrals. For example, take the following relation, where I is the identity matrix:. Doing the wave Take a look at the position operator, R, in a continuous basis. Applying this operator gives you r, the position vector:.

In this equation, applying the position operator to a state vector returns the locations, r, that a particle may be found at. You can expand any ket in the position basis like this:.

The quantum physics in the rest of the book is largely about solving this dif- ferential equation for a variety of potentials, V r. For example, you may have a particle trapped in a square well, which is much like having a pea in a box. Or you may have a particle in harmonic oscillation. Quantum physics is expert at handling those kinds of situations.

Chapter 3. Stuck in an energy well? Go get help! In this chap- ter, you get to see quantum physics at work, solving problems in one dimension. You see particles trapped in potential wells and solve for the allowable energy states using quantum physics.

But as you know, when the world gets microscopic, quan- tum physics takes over. Looking into a Square Well A square well is a potential that is, a potential energy well that forms a square shape, as you can see in Figure Figure A square x well. Using square wells, you can trap particles.

Therefore, the particle has to move inside the square well. So does the particle just sort of roll around on the bottom of the square well?

Not exactly. The particle is in a bound state, and its wave function depends on its energy. The energy of the allowable bound states are given by the following equation:. Figure A potential well.

In this section, you take a look at the various possible states that a par- ticle with energy E can take in the potential given by Figure Quantum- mechanically speaking, those states are of two kinds — bound and unbound. This section looks at them in overview. In other words, the particle is confined to the potential well.

A particle traveling in the potential well you see in Figure is bound if its energy, E, is less than both V1 and V2. In that case, the particle moves in a classical approximation between x1 and x2.

Bound states are discrete — that is, they form an energy spectrum of discrete energy levels. This section looks at them separately. The energy eigenvalues are not degenerate — that is, no two energy eigenvalues are the same see Chapter 2 for more on eigenvalues. The energy spectrum is continuous and the wave function turns out to be a sum of a function moving to the right and one moving to the left.

The energy levels of the allowed spectrum are therefore doubly degenerate. Trapping Particles in Infinite Square Potential Wells Infinite square wells, in which the walls go to infinity, are a favorite in physics problems.

You explore the quantum physics take on these problems in this section. Finding a wave-function equation Take a look at the infinite square well that appears back in Figure So now you have a second-order differential equation to solve for the wave function of a particle trapped in an infinite square well. You get two independent solutions because this equation is a second-order differential equation:. What are the boundary conditions? The wave function must disappear at the boundaries of an infinite square well, so.

This is the lowest physical state that the particles can occupy. Just for kicks, put some numbers into this, assuming that you have an electron, mass 9. Normalizing the wave function Okay, you have this for the wave equation for a particle in an infinite square well:. You can see the first two wave functions plotted in Figure Figure Wave functions in a square well. Adding time dependence to wave functions Now how about seeing how the wave function for a particle in an infinite square well evolves with time?

In fact, because E is constant, you can rewrite the equation as. In this case, the solution is. Chapter 3: Getting Stuck in Energy Wells 67 So when you add in the time-dependent part to the time-independent wave function, you get the time-dependent wave function, which looks like this:. Shifting to symmetric square well potentials The standard infinite square well looks like this:.

But what if you want to shift things so that the square well is symmetric around the origin instead? So as you can see, the result is a mix of sines and cosines. The bound states are these, in increasing quantum order:. Limited Potential: Taking a Look at Particles and Potential Steps Truly infinite potentials which I discuss in the previous sections are hard to come by. In this section, you look at some real-world examples, where the potential is set to some finite V0, not infinity.

For example, take a look at the situation in Figure There, a particle is traveling toward a potential step. There are two cases to look at here in terms of E, the energy of the particle:. In this equation,. In other words, k is going to vary by region, as you see in Figure Treating the first equation as a second-order differential equation, you can see that the most general solution is the following:. Chapter 3: Getting Stuck in Energy Wells 71 What this solution means is that waves can hit the potential step from the left and be either transmitted or reflected.

Given that way of looking at the problem, you may note that the wave can be reflected only going to the right, not to the left, so D must equal zero. That makes the wave equation become the following:. The term Aeik1x represents the incident wave, Be—ik1x is the reflected wave, and Ceik2x is the transmitted wave. Calculating the probability of reflection or transmission You can calculate the probability that the particle will be reflected or trans- mitted through the potential step by calculating the reflection and transmis- sion coefficients.

These are defined in terms of something called the current density J x ; this is given in terms of the wave function by. If Jr is the reflected current density, and Ji, is the incident current density, then R, the reflection coefficient is. You now have to calculate Jr , Ji, and Jt. And this just equals.

In other words,. In particular,. So already you have a result that differs from the classical — the particle can be reflected at the potential step. See what quantum phys- ics has to say about it. But hang on; E — V0 is less than zero, which would make k imaginary, which is impossible physically. There are two linearly independent solutions:. That means that you have a complete reflection, just as in the classical solution.

You can use the continuity conditions to solve for C in terms of A:. That brings you to potential barriers, which I discuss in the next section.

Hitting the Wall: Particles and Potential Barriers What if the particle could work its way through a potential step — that is, the step was of limited extent?

You use the continuity condi- tions, which work out here to be the following:. So putting all of these equations together, you get this for the coefficient E in terms of A:. But now E — V0 is less than 0, which would make k imaginary. And use this for k Finding the reflection and transmission coefficients How about the reflection and transmission coefficients, R and T? Tunneling is possible because in quantum mechanics, particles show wave properties. Tunneling is one of the most exciting results of quantum physics — it means that particles can actually get through classically forbidden regions because of the spread in their wave functions.

Among other effects, tunneling makes transistors and integrated circuits possible. You can calculate the transmission coefficient, which tells you the prob- ability that a particle gets through, given a certain incident intensity, when tunneling is involved. Read on. The result of the WKB approximation is that the transmission coefficient for an arbitrary potential, V x , for a particle of mass m and energy E is given by this expression that is, as long as V x is a smooth, slowly varying function :.

So now you can amaze your friends by calculating the probability that a par- ticle will tunnel through an arbitrary potential.

There are plenty of particles that act freely in the universe, and quantum physics has something to say about them. If you add time-dependence to the equation, you get this time-dependent wave function:. The probability density for the position of the particle is uniform throughout all x!

So what that equation says is that you know E and p exactly. And if you know p and E exactly, that causes a large uncertainty in x and t — in fact, x and t are com- pletely uncertain. Trying to normalize the first term, for example, gives you this integral:. That looks pretty circular. Choose a so-called Gaussian wave packet, which you can see in Figure — localized in one place, zero in the others.

Figure A Gaussian wave packet. The probability is. H armonic oscillators are physics setups with periodic motion, such as things bouncing on springs or tick-tocking on pendulums.

There are many, many physical cases that can be approximated by harmonic oscillators, such as atoms in a crystal structure. In this chapter, you see both exact solutions to harmonic oscillator problems as well as computational methods for solving them.

Founding Father Alexander Hamilton. The Hamiltonian will let you find the energy levels of a system. The key point here is that the restoring force on whatever is in harmonic motion is proportional to its displacement. Understanding total energy in quantum oscillation Now look at harmonic oscillators in quantum physics terms.

The Hamiltonian H is the sum of kinetic and potential energies — the total energy of the system:. The problem now becomes one of finding the eigenstates and eigenvalues. Unlike the potentials V x covered in Chapter 3, V x for a harmonic oscillator is more complex, depending as it does on x2. So you have to be clever. The way you solve harmonic oscillator problems in quantum physics is with operator algebra — that is, you introduce a new set of operators.

You use the creation and annihilation operators to solve harmonic oscillator problems because doing so is a clever way of handling the tougher Hamiltonian equation see the preceding section. These operators make it easier to solve for the energy spectrum without a lot of work solving for the actual eigenstates. In other words, you can understand the whole energy spectrum by looking at the energy difference between eigenstates. First, you intro- duce two new operators, p and q, which are dimensionless; they relate to the P momentum and X position operators this way:.

The N operator returns the number of the energy level of the harmonic oscil- lator. The Schrodinger wave equation for a free particle is given by. Therefore energy of the free particle is not quantised. Open navigation menu. Close suggestions Search Search. User Settings. Skip carousel. Carousel Previous. Carousel Next. What is Scribd? Explore Ebooks. Bestsellers Editors' Picks All Ebooks. Explore Audiobooks.

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