What is the Ultraviolet Catastrophe?

Towards the end of the 19th century, physics was flourishing, the principles of thermodynamics had helped fuel the industrial revolution, and Faraday had recently discovered how to generate electricity. Furthermore, James Clark Maxwell's great unification of electricity and magnetism had led to the surprising realization that light was an electromagnetic wave driven by the power and scope of these new theories. Physicists began using them to address some of the great unsolved problems of the time. One of those problems involved explaining how hot objects emit light. It was well known that if you heat a lump. of the metal, it will first shine dull red, then orange-red, then yellow, and finally bluish-white, at the same time becoming brighter.

Physicists wanted to explain how the intensity or brightness of the emitted radiation is related to the wavelength or color. They realize that this would involve understanding. how the chunk of metal interacts with radiation, the light emitted on a fundamental level; However, when Maxwell's theory was used in conjunction with classical thermodynamics to calculate the emission profile of a hot object, he predicted that all such objects should emit an infinite amount of energy. at small wavelengths or high frequencies because high frequency light was called

ultraviolet

light and because this prediction was clearly problematic, this problem became known as the

ultraviolet

catastrophe

and represented a complete failure of classical physics and went to the Trying to solve this problem forced Max Planck to introduce the revolutionary concept of energy quantization, the idea that energy is transferred discretely in chunks rather than continuously.

After Planck's work on this topic, the world would never be again. the same in order to properly understand Plank's solution to the problem and why this represents such an important turning point in the history of science, we must first understand the cause of the problem and to understand the cause we must start with some basic concepts when you illuminate an object, some of it can be reflected, some of it can be transmitted and some of it can be absorbed. Darker objects tend to absorb more light that falls on them and reflect less than lighter objects, which is why they appear dark in the first place, but also why they tend to heat up more quickly.

Think about

what

black surfaces are like. It gets hot in summer compared to white surfaces. Physicists refer to an object that absorbs all light incident on it as a blackbody for obvious reasons if the temperature of a blackbody is lower than the temperature of the environment in which it is located. placed, it will continue to absorb energy from the environment until it has reached the same temperature, at which stage it is said to be in thermal equilibrium; However, if the blackbody has a temperature that is higher than that of its surroundings, then it will radiate energy until it has lowered its temperature enough to reach thermal equilibrium, therefore, a blackbody is not only a perfect absorber of radiation but also a perfect radiation emitter.

Furthermore, the experiment shows that the wavelength intensity profile of a black body has a universal character that does not depend on the shape or composition of the black body but only on its temperature. It is this relative simplicity that makes the black body very attractive to physicists and the logical starting point for any investigation into the radiation properties of hot objects. An example of a real black body studied by physicists is called a jeans cube in honor of the physicist Sir James Jeans who together with Lord Rayleigh provided one of the first attempts to explain the radiation emitted by a black body.

A jeans cube can be considered as a cubic metal box with a very small size. hole on one side the radiation incident on the hole from the outside will enter the cube and be reflected back and forth by the walls until it is absorbed, since the hole is very small, only a small fraction of the Radiation inside the cube will escape. Back through the hole, therefore, from the hole's perspective, essentially all radiation incident on it will be absorbed and therefore the hole acts as a blackbody. We can also consider the reverse process in which the walls of the cube are heated uniformly to a temperature such that the walls begin to emit radiation and fill the cube since a small fraction of this radiation will then escape through the hole, we see that the hole acts as a radiation emitter since we have already established by the absorption properties of the hole that it behaves as if it is a black body, it follows that the radiation emitted from the hole must have the intensity profile of a black body, in addition Since the hole is simply sampling the thermal radiation inside the cube, it is clear that the radiation inside the cube must also have the wavelength.

Intensity profile of a black body Groundbreaking experimental work on the radiation properties of idealized black bodies was carried out by Otto Lummer and Ernst Pringsheim and these experiments provided valuable data for theoretical physicists to investigate experimental work involving measuring radiation. spectral radiance of the black body as a function of wavelength, this resulted in characteristic curves that varied with temperature, the lower the temperature, the flatter the curve, spectral radiance is defined as the energy per second per unit of area per unit wavelength, which is simply the intensity per unit wavelength and, since intensity is defined as power per unit area, it follows that the units of spectral radiance are watts per cubic meter to find the intensity from one of these graphs you simply have to find the area under the curve in other words, the intensity is the integral of the spectral radiance with With respect to wavelength experiments like these, they reveal patterns in the data, in particular emerged two surprising empirical laws.

The first of them was Stefan Boltzmann's law, which related the intensity of the emitted radiation to the temperature. You can clearly see from the graph that as the temperature increases. The area under the corresponding curve increases and therefore the intensity increases. Stefan Boltzmann's law quantifies this relationship and says that the intensity is proportional to the fourth power of the temperature. The second empirical law is the v in the displacement law that relates the wavelength corresponding to the peak the spectral radiation with temperature as the temperature increases the wavelength corresponding to the maximum emission is reduced This fits with

what

we have already seen when heating a piece of metal as the temperature increases of the shiny metal its color changes from red to orange and then to yellow, which corresponds to increasingly shorter wavelengths of light and this is codified in the vian shift law, although Stephen Boltzmann's law and the law of vian displacement provided valuable information about blackbody radiation.

Theoretical physicists wanted more, firstly they wanted to know what the spectral radiation function was and secondly they wanted to be able to derive an expression for spectral radiation using the known laws of physics and this is what Rayleigh and Jeans did . They turned their attention to the theoretical study of radiation within the gene cube and applied the best theories of the time. To address this problem, what were the known laws of physics? Since Rayleigh and Jeans wanted to study the radiation properties of a black body, it made sense for them to use the successful laws of electromagnetism and in particular the recently discovered properties of electromagnetic radiation as elucidated by James Clark Maxwell according to Maxwell light consists of a self-propagating electromagnetic wave that travels at the speed of light through a vacuum Raelian genes also resorted to the statistical laws of thermodynamics proposed by Ludwig Boltzmann according to Boltzmann the temperature of an object is simply a statistical average of the energy of the particles that make up the object armed with Maxwell's electromagnetic waves and Boltzmann's statistical thermodynamics.

Rayleigh and Genes were confident that they could successfully derive the spectral radiation function for a blackbody. They just needed a clear strategy in the first place. I realized that it would be more useful and simpler if they specified the radiation spectrum within the gene cube in terms of energy density rather than spectral radiance, so here's what we're going to do: Energy density is simply the energy contained in a unit volume of the cube at a temperature t in the frequency interval between f and f plus df Once the energy density has been established, the spectral radiance can be calculated very simply since the spectral radians and the energy density are proportional to each other, in fact the radian spectral radiation is equal to c over four pi times the energy density and we will return to this result later, okay, but how do you find the energy density?

Rayleigh and Jeans' approach was three-fold. First, you must count the number of electromagnetic waves. that fit inside the cube, secondly you need to calculate the average energy of these waves when they are in thermal equilibrium and finally you combine these two results to find the energy density for a particular range of frequencies, so let's do this to start with, we need to understand how to mathematically describe the waves, since we are going to describe the radiation inside the cube in terms of electromagnetic radiation which, as already mentioned, consists of waves, we can describe a wave traveling to the right using the following sine function here and represent the displacement of the wave for a given x coordinate and a given time coordinate, since the largest possible value of the sine function is 1, we see that uppercase a represents the amplitude and this is the maximum possible displacement, so if, for For example, if it were a wave on a string, a would represent the maximum height. that the string moves away from the equilibrium position the minus sign tells us that the wave is moving to the right and if we let time flow we can see this the constant k is known as the wave number and is defined as 2 pi over Lambda represents the spatial frequency of the wave, on the other hand, omega is simply proportional to the frequency of the wave and that is the temporal frequency, that is, the number of complete waves that pass a point every second.

Now we can also write an expression that describes a moving wave. to the left and you see that the minus sign has been replaced by a plus and this causes the wave to move to the left, so what would happen if we had a wave that was moving to the right and it encountered a wave that was moving to the right? moves to the left in the same region of space? In that case we would refer to the principle of superposition which states that the net displacement at a place in space is simply the sum of the individual displacements at that point, so if we wanted to find the net displacement due to our wave moving towards For the right and leftward wave passing through the same point in space, we simply add the individual displacements.

Now we can simplify this expression by using some trigonometric identities, in particular the sine of a plus b and the sine of a minus b, if we add these two trigonometric identities, we find this. purple expression sine of a minus b plus sine of a plus b is equal to 2 sine a cos b now if we pair terms we see that a can be identified with kx and b can be identified with omega t and therefore the net displacement can be written as 2a sine kx cos omega t so what does this look like? Well, we can visualize it with the following animation, you will see that the left and right moving waves add up to form the green wave and if we eliminate the left and right moving waves and only focus on the net displacement, we get the following pattern of standing wave.

It is called a standing wave pattern for obvious reasons because the pattern does not move left or right, it only moves up and down. So how is this related to the electromagnetic radiation inside the gene cube? Well, suppose the gene cube is aligned along an x,y,z axis and the length of each side of the cube is l, then the electromagnetic radiation that is reflected back and forth can be analyzed into three components , consider the x component, all the radiation along this component that falls on the wall is reflected by it and the incident and reflected waves combine to form a standing wave along the x direction, since electromagnetic radiation is a transverse vibration with the electric field oscillating perpendicular to the direction of propagation and since the propagation direction is perpendicular to the wall, it follows that the electric field is parallel to the wall, however, a metallic wall cannotsupport an electric field parallel to the surface, since electric charges can always flow in such a way as to neutralize the electric field, therefore the electric field must be zero on the walls of the container which is the standing wave associated with the component x of the radiation must have a node also known as zero amplitude where x is equal to zero and x is equal to l these conditions put a limit on the possible wavelengths and therefore on the frequencies of the em radiation within the cube Clearly the same also applies to the y and z directions.

To see how this works mathematically, we first need to determine an expression for the pattern of the stationary electric field inside the cube, but this is easy because we have already seen that a wave pattern is described by the next function and now we just need to replace the displacement y with the electric field intensity e. Now we can apply the boundary conditions to our standing wave function, for example, we know that when x is equal to zero, we are located on the side of the cube and therefore the electric field intensity must be equal to zero why well For the case x is equal to zero we see that this is true because the sine of zero is equal to zero, however we also know that when x is equal to l we are on the edge of the cube and therefore the field intensity must also be equal to zero now if we set x equal to l in the stationary field function we see that this will only be zero if the sine of k l is equal to zero and that this will be true whenever kl is an integer multiple of pi radians. equal to n pi over l now if we combine this with the definition of k that I mentioned earlier, k is equal to 2 pi over lambda, where lambda is the wavelength, then you can see that we can rewrite the wavelength as 2l over n where n is an integer one two three, what does this tell us?

It tells us that the wavelength can only take particular values ​​now if we rewrite this in terms of the frequency we see that the frequency of the wave can also only take certain integer values ​​so let's see what this looks like so imagine that We are focusing only on the x direction, here are the two sides of the cube separated by a distance l and we know that the wavelength is given by two l over n, where n is an integer so when n is equal to one we see that the wavelength is given by two l What happens when n is equal to 2?

Well, in that case the wavelength is given by l and we can continue with n equals 3 n equals 4 n equals five and n equals six so we see that only certain wavelengths can fit inside the cavity or cube. Now we also know that we can write an expression for the frequency, so by adding the frequencies for each of these cases we see that the frequencies also consist of certain integer values ​​that we can. represent these allowed values ​​of frequency in terms of a diagram consisting of an axis on which we plot a point at each integral value of n.

Such a diagram allows us to count the number of states allowed in any given frequency range, for example we can specify the distance d from the origin as 2 l over c times the frequency, then we can calculate the number of states allowed in the frequency range f to f plus df by observing the difference in distance from the origin, since the points are uniformly distributed along the n axis it is evident that the number of points falling between the two limits will be proportional to df but will not actually depend on f . We can see this explicitly by calculating nf df which represents the number of states in the interval df now this is simply 2l over c f plus df minus 2l over cf which is equal to 2l over c times df now we need to multiply this by an additional factor of 2 since that for each of the allowed frequencies there are actually two independent waves corresponding to the two possible polarization states. of the electromagnetic wave, so our final answer for the density of states is nf df is equal to four l over c multiplied by df.

Now this example gives us a framework for how we might calculate the number of standing waves allowed in the three-dimensional gene cube, but Before we get ahead of ourselves, let's first look at a simple generalization to two dimensions. Let's now focus on the x and y dimensions of our cube. We will consider radiation of wavelength lambda propagating in a direction defined by the two angles alpha and beta as seen in In the diagram, radiation in this arbitrary direction must form a standing wave, since we have already seen that the components x and y form standing waves.

The diagram indicates the locations of some of the fixed nodes of this standing wave by a set of lines perpendicular to the direction of propagation. The distance between these nodal lines of radiation is half a wavelength, as is the case with all standing wave patterns. . We have also indicated the locations on the two axes of the nodes of the x and y components, as well as the corresponding wavelengths along the x and y axes. Looking at the geometry of this diagram we see that lambda over 2 is equal to lambda x over 2 cos alpha which we can simply write as lambda equals lambda x cos alpha rearranging this expression we find that lambda is equal to lambda over cos beta now we can use the relation we derived earlier which links the wavelength to the case length and the integer n specifies this in the x direction we find that lambda x is equal to 2l over nx and this means that we can write n over nz and therefore nz is equal to 2l cos gamma over lambda where gamma would be the angle between the normal wave and the z axis.

If we now consider nx squared plus ny squared plus nz squared we obtain the following expression 2l over lambda all squared multiplied by this expression inside the parentheses, but we know that cos squared alpha plus cos squared beta plus cos squared gamma equals one, this is just the three-dimensional version of the Pythagorean theorem, so nx squared plus ny squared plus nz squared is simply equal to two l over lambda all squared if we rearrange this we find that lambda is equal to two l divided by the square root of nx squared plus ny squared plus nz squared and now if we express this in terms of frequency using the fact that f is equal to c over lambda we see that f is equal to c over 2l multiplied by the square root of nx squared plus ny squared plus nz squared now we want to use this expression to determine the density of states in our three-dimensional cube we will follow the same logic as with the one-dimensional example we considered above in that we will attempt to count the number of states in a particular frequency range by considering axes along which we label the integer values ​​from n to We simplify our calculation of density of states, we will consider a sphere with its center located at the origin of the nx, ny and nz axes, we will focus only on the positive values ​​of x, y and z, which represent 1 8 the volume of the entire sphere and then use the symmetry to complete the picture, we will also consider a two-dimensional cross section of the octant.

Now let's look at this in a little more detail. If we assume that our sphere has radius r, then we can determine the density of states by considering how many states there are between radius r and radius r plus dr where dr is a segment of small radius we can express the radius of our sphere in terms of nx n and y nz using the following expression, we can then clearly combine this with the expression we derived earlier for the frequency and see that the radius can be written in terms of the frequency as r is equal to 2lf over c, if we then differentiate this expression with respect to f We find that dr times df is equal to 2l over c and therefore dr is equal to 2l over c multiplied by df so what is nrdr?

Well, we see in our diagram that this will simply be equal to the volume of the small segment labeled dr, which for the entire sphere would be equal to the surface area of ​​the sphere 4 pi r squared multiplied by segment length dr however, because Since we are only dealing with 1 8 the volume of the entire sphere, we need to multiply our expression by 1 8 and so we get nrdr is equal to 1 8 times 4 pi r squared times dr and this simplifies to pi r squared dr over 2. We can rewrite this expression in terms of frequency instead of radius using the relations we derived previously, we find that nfdf equals pi over 2 times 2l over c o f cubed squared gl and this simplifies to nf df equals 4 pi l cubed over c cubed f squared df again we need to multiply this expression by 2 to account for the two possible polarization states of the electromagnetic radiation and this now represents our final expression for the number of states that exist within a particular frequency range. that the number scales with f squared, indicating that the number of states grows rapidly at high frequency or short wavelength.

Now that we have an expression that allows us to count the number of wave frequencies allowed within the cube, our next step is to calculate the average energy. of each of these waves and this is what we turn to now to calculate the energy of each wave. Rayleigh and the genes were based on a well-established result from the classical theory of thermodynamics called theequipartition theorem according to this theorem if you have a gas in thermal equilibrium comprising billions and billions of small particles in random motion, then the thermal energy available within the system will be distributed, on average, equally among the particles within the gas.

Consider, for example, a box containing a monatomic gas in which each molecule consists of a single atom. In classical physics, each atom could be thought of as a small sphere that moves randomly within the box and collides with others. atoms on the walls of the box. In this case, the kinetic energy of each atom can be divided into three modes corresponding to motion in the x, y and z directions, given the incredible number of atoms within a gas, on average the x and z values ​​for the kinetic energy are the same ; In other words, if you were to add x joules of energy to the box, then x over 3 joules will appear as increase in kinetic energy in each of the three coordinate directions for a system in thermal equilibrium each mode or degree of freedom has an average energy equal to half kt where k refers to the Boltzmann constant.

You can think of the Boltzmann constant as representing the exchange rate between currencies. of energy and temperature, for example, a monatomic gas that has three possible degrees of freedom corresponding to motion in each of the three coordinate directions has an average energy of three times half a kt or three over two kt to find the total energy of the gas . then you would simply need to multiply the average energy of each particle by the number of particles, this gives in our example the average energy multiplied by n. So how does this apply to waves? Raelian genes.

Reasons why the same statistical law should be valid for waves inside the cube. This is because the law of energy equipotentialization was believed to apply to any classical system containing a large number of entities of the same type in equilibrium. For the case at hand, these entities are standing waves inside the cube, each with a degree of freedom. corresponding to their electric field amplitudes, therefore, on average, all their kinetic energies have the same value, half a kt. However, each sinusoidally oscillating standing wave has a total energy that is twice its average kinetic energy; In other words, the total energy is actually kt.

This is common property. of physical systems that have a single degree of freedom and that execute simple harmonic oscillations in time, it is worth pausing for a moment to consider where this result originates from, this will be particularly important later when we try to locate the cause of the ultraviolet

catastrophe

, the fact that the average energy is equal to kt has its origin in a more complete result of classical statistical mechanics called the Boltzmann distribution. The Boltzmann distribution function provides complete information about the energies of the entities in our system, including of course the average value of the energies which can be calculated using the following expression, although this expression seems quite alarming, it is actually quite simple, the integrand in the numerator can be considered as the energy weighted by the probability that the entity encounters this energy by integrating over all possible energies the average is obtained the value of the energy, on the other hand, the denominator can be considered represents the probability of finding the entity with any energy, now the integral in the denominator is simple and isYou may find that it is equal to kt, the integral in the numerator requires a little more work, however, we can use integration by parts by noting that it is possible to use the product rule to write the following expression, if we then integrate both sides , we see that this is equal to the following expression, a slight rearrangement of this allows us to integrate the numerator and we find that the result is equal to kt squared, so putting everything together we find that the average energy is equal to kt squared divided by kt, which is simply kt as expected, we are finally in a position to combine our results.

We have seen that the number of waves in our cube can be calculated using the density of states function and we have just seen that the average energy of each wave is given by theequipartition theorem. Now if we want to calculate the energy density inside the cube. cube then we need to calculate the energy per unit volume. It is clear that the total energy can be calculated by multiplying the density of states by the average energy per state and then we can divide by the volume of the cube to find the energy density. The expression can be written as rho of fdf where rho represents the energy density and df the frequency range under investigation.

We can simplify this expression and arrive at our final answer for the energy density expressed in terms of frequency, but what if we want to express the energy density in terms of wavelength in that case we can relate the energy density in terms of wavelength with the energy density in terms of frequency using the following expression where in this expression both rho of f and rho of lambda are positive, the minus sign originates from the fact that df is equal to minus d lambda and why this is true. This is true because as the frequency of a wave increases, the wavelength decreases and this is indicated by the minus sign.

Now we know that the frequency is given by the speed. of light divided by the wavelength of an electromagnetic wave and therefore we can write df times d lambda equals minus c over lambda squared and then this allows us to write df equals minus c over lambda squared d lambda putting together all this we can write rho of lambda d lambda is equal to minus 8 pi f kt squared over c cubed df putting the expressions that we have just derived we see that this is simplified and we can write the final answer as rho of lambda d lambda is equal a 8 pi k t over lambda to lambda 4 d and this represents the energy density in terms of the wavelength, so what does this function look like?

If we plot the energy density on the y-axis and the wavelength on the x-axis, then we know what we would like the function to be. It seems because we have seen in the experiment the following smooth curve, however, when Rayleigh and Jeans plotted their function they found the following alarming result: we see that at small wavelengths the energy density increases rapidly. This rapid increase can be seen in the energy density function. We see that when lambda is small then lambda raised to the power of 4 will be small and therefore the energy density will be huge. If we think about this in terms of frequency, then a small wavelength corresponds to a high frequency and as there is no upper limit to the frequency within our cube as the frequency goes to infinity so does the energy density and it is This catastrophic result is known as the ultraviolet catastrophe.

The Raelian genes had shown that the best understood classical laws of physics were incapable of describing even the most common phenomena, namely the radiation properties of hot objects, but not everything was lost from the ashes of classical physics, a new revolutionary framework would emerge that would not only change the laws of physics but also life on Earth. Max Planck was a modest German physicist who dedicated most of his life to the study of thermodynamics. In a now famous lecture delivered on December 14, 1900, Planck announced that the conclusions obtained by Rayleigh and Jeans could be remedied and the danger of the ultraviolet catastrophe if it is postulated that energy can only exist in the form of certain discrete packets called quanta.

To reach this conclusion, Plank had worked tirelessly on the problem for several years, sometimes desperate at the difficulties he encountered, Planck decided to focus on the experimental results and compare them with the Rayleigh experiment. The gene curve, although the Rayleigh gene result decomposes at small wavelengths, actually matches the experimental data quite well for long wavelengths, suggesting that the average energy may be equal to kt in the long wavelength limit; On the other hand, it is clear from the experiment. data that the average energy should tend to zero for short wavelengths or high frequencies Planck concluded from the data that the average energy of each wave should depend on the frequency of the wave and therefore the equipotentialization of energy.

Planck's great contribution came when he realized that he could obtain the required energy limit if he modified the calculation of the Boltzmann distribution of the average energy. He did this by treating energy as if it were a discrete variable rather than the continuous variable of classical physics in particular. He imagined that energy could only take on certain discrete values ​​and that the total energy of an object would be an integer multiple of a basic amount of energy, so what effect does this have? It turns out that it has a profound effect on the average energy calculated using the Boltzmann distribution in In particular, the change from continuous to discrete energy corresponds to the replacement of integrals with sums in the average energy equation.

So how do we evaluate this expression well? The addition in the denominator is relatively simple as we see that it is simply a geometric addition, so we can write this as 1 over 1 minus e a minus epsilon over kt the numerator requires a little more work. Note that we can implement a neat trick and rewrite the sum in differential form where we have defined f of epsilon as the same sum that appears in the denominator of the average energy expression, if we then calculate df times d epsilon we find the following rather gruesome expression , putting it all together, we find the following expression for the average energy, so what does it all mean?

First, note that when epsilon is much smaller than kt e al epsilon over kt is very small, so we can approximate the exponential by the first few terms of the Taylor series expansion of the exponential. This shows us that in the small epsilon limit the average energy is simply equal to kt in the On the other hand we see that in the large epsilon limit the exponential in the denominator tends to infinity faster than epsilon in the numerator and therefore the average energy tends to zero now as we have already seen in an analysis of the experimental data the average energy tends to kt when the frequency tends to zero and the average energy tends to zero when the frequency tends to infinity and we have just seen that since A theoretical analysis the average energy also tends to kt as epsilon tends to zero while the average energy tends to zero as epsilon tends to infinity, so by comparing these two sets of observations, Planck realized that, whatever For it to be epsilon, it must be an increasing function of frequency.

Planck's detailed numerical work led him to propose that the energy epsilon is proportional to the frequency which he then introduced a proportionality constant so he could write epsilon equal to hf and he was finally able to rewrite the average energy as hf over e to hf over kt minus one armed With this new calculation of the average energy Planck was able to combine this result with the state density function to derive a new expression for the energy density of a blackbody radiator, rewriting this in terms of wavelength, he derived the following function of energy density, so what does this look like when Plank plotted his new energy density function and found a remarkable match?

Furthermore, with the experimental data, its function had the additional advantage of reducing to the energy density function of the Rayleigh gene in the long wavelength limit, as can be seen by looking at the first terms of the series expansion Taylor of the exponential that appears in the detailed denominator. Experimental work done in 1916 by William Koblenz provided wonderful confirmation of the Planck energy density function. The solid line in this graph represents the theoretically predicted Planck curve and the circles represent the experimental results. Koblenz used his results to determine a value for Planck's constant h that is remarkably close to the value accepted today.

Although Plank's work is based on the radiation properties of hot objects, his quantum hypothesis would point to the birth of a whole new branch of physics, in particular its understanding that any physical system undergoing simple harmonic motion can only possess energy that satisfies the ratio and equal. nhf would revolutionize the way physicists describe the structure of matter by visualizing this. Consider the energies allowed in a classical system that oscillates sinusoidally with frequency f, in this case the energy is distributed continuously as can be seen in the diagram, while the energies allowed according to Planck's theory. The postulates are discretely distributed since they can only assume the values ​​nh if we say that the energy is quantized, with n being the quantum number of a permitted quantum state.

Thirteen years later, Niels Bohr would use Planck's energy quantization hypothesis to develop his model. energy level of the atom. and launched the field of quantum mechanics many years later, Planck described his mental state during those productive years as follows. I will briefly summarize what I did. It can simply be described as an act of desperation in nature. I have a peaceful inclination and reject all dubious adventures, but by then I had been struggling unsuccessfully for six years with the problem of the balance between radiation and matter and I knew that this problem was of fundamental importance for physics, so it had to be found a theoretical interpretation at any price, no matter how high.