4. Solutions to Schrödinger Equation, Energy Quantization

the following content is provided under a Creative Commons license your support will help MIT OpenCourseWare continue to offer high quality educational resources for free to make a donation or view additional materials from hundreds of MIT courses visit MIT opencourseware at ocw.mit.edu we recap of the last lecture what we talked about is the wave ledger we particle duality starting Punk Einstein relations energy is related frequency momentum is ready to the wavelength of the particle and that laid the foundation for the wave particle duality although Einstein focused on his effort on the photons in the hung this was extended to they see the mature waves and for mature waves will have different ways to describe it their different approaches matrix approach the operator approach but the one that I use here is the Schrodinger equation that says they describe this wave function which is the function of time and space and the we function on the left-hand side essentially this is the like at your kinetic energy this is a potential energy and the right-hand side is a time dependent term and the solution as we'll see in today's lecture of this equation will give you a wave type of solution and uh what's interesting is the meaning of this wave function was not clear at the time that this equation was written down so the it turns out that the explenation correct explanation of the wave function we function itself is a complex number doesn't have physical meaning but the we function times complex conjugate will give a real library and this gives the probability of finding this material wave a certain time and location so this is the explanation given by born and that create a lot of trouble because that means is uncertain in the quantum world is probability so there is an expectation value they the mean value and there is also standard deviation and that standard deviation net who the Heisenberg uncertainty principle we commented in the last lecture and then we did a little bit maths we say okay let's do a separation of variable because this will be function is the function of time and space as long as this u potential energy the constraint this material wave so the equation is if you look at it so the first term is the kinetic energy and second term is the potential energy and what's differ for each problem is really this potential energy that constraints are different so if u is not dependent on time then we can do separation variable to separate the wave function into a space part and the time part time part was easy you can easily solve it and turns out that separation of variable constant ye has a physical meaning and that minute is the energy of the system and with that we substitute into from this time-dependent equation we can get the steady state equation where the steady state is space dependent part of the wave function can be determined and you will see that this is actually a eigenvalue equation it's here here on the right hand side zero so what you actually use is not only find the wave function but also find the eigen value of the energy of the system so this will put a constraint on the possible energy of the system that's where the energy quantization word naturally came up so this is a where we were in the last lecture and today we will use this equation and give a few simple solutions so today what we'll go in first I will give you example solutions and if you recall in the first two lectures we talked about what's the energy alpha or molecule right we have kinetic energy we have vibrational energy rotational energy electronic energy and I'm going to use the quantum mechanics roading equation to see what are they storing go to in the quantum mechanics impose on this energy forms and the quantization and then from this we will see the important one some important concept what is the quantum state and what are they degenerate sees and if you recall that we talked about the the counting right we see okay how many particles we have and this degeneracy is an important step in doing the correct accounting when we later on to the transport problem so account of how much energy the system really has so let's start with the example solutions and we said the difference for each problem is really that potential you'll write the first part is the kinetic energy is say once you know the mass that's the term you have and each problem each particle is constrained by different potentials right our air molecule in this room electron in a film or the vibration energy those are different potential constraints right so let's look hide up first the simplest one there is no potential no constraint a free particle a complete free and let's look at 1d so we don't need to write the three dimension so 1d free particle what does that tell you so ah we have already the time dependent term right and so we only need to go to find out from the steady state the space dependent term by combined together we know what's the wave function and if it's free there's no constraint this you the potential energy is zero everywhere and so my steady state equation if you copy that into 1d is mass h-bar squared to M D psi DX minus r e+ i equals zero e is the energy of this particle the eigen value of the problem and this equation i think you can all solve it let's go one more step I put them together this constants so what I have is a K square plus I equal to zero well this K square is 2m at e / H bar square right it's just a location for my cake and I solve this equation second order differential equation with constant coefficient K not a space dependent term right that's the easy one so you find the card ristic root of this terrific equation that's plus minus K so what we have is then plus I X is a Ematic I KX plus b e plus ikx do you agree that's the solution right now we combine this space dependent solution with the time dependent solution - ie H bar I say that's a time dependent right and ay yi I say it's equal si is the energy e equals H nu so I could write this right so the wave function as a function of time and space could be then written as the explanation mass I Omega T up plus KX plus B exponential minus I Omega t minus KX well this e equals H bar Omega H bar is H over 2 pi so this is the angular frequency rather than nu nu Omega is 2 pi nu right so that's the solution for this pre-packed a free particle in space from minus infinity to plus infinity what you can say is this solution tells you the particle is away this is tip your wave right this is a wave propagate into which the ration negative with the direction X direction right omega t minus KX here is omega t plus KX so you have to counter propagating waves in free space and let's go from minus infinity to plus infinity particular is everywhere right this is the uncertainty for pre particle you can pinpoint where it is okay and so that's the free particle and basic it tells us it's a wave and if you look at this K is the wave vector right if you look at this wave case weight vector and if I go to combine and write a energy what's the energy of this particle is H bar square K square divided by 2m right here this is my definition 4k K is the wave vector okay I write the energy H Bar K go back to the Einstein relation there are P equals H lambda is h-bar K that's the momentum so it comes back to essentially the p square 2 m that's the kinetic energy free particle right kinetic energy momentum energy relation energy equals one-half MV square right so this is the 1d free particle and now let's put a constraint on it right so ah 1d quantum well and we already went to the web and look at the people actually working on Commonwealth so the deposits in films and in that case your electron in the film is subject to the constraint because outside the film is a vacuum or another material right so now we give the simplest potential form for a say constraint simcha simple constraint so here is the potential let's say this is the day house this is the film if I deposit a solid film and this is the thickness is 0 to D right and potential let's say I take this as 0 this as the infinite right so that's a simplification in the real tight real case vacuuming may not be the difference between vacuum and the film the energy is a pretty much the work function of the electron we said before right that's a few electron volt so it's not a 0 to infinite but ah for simplicity of my solution here I say here is 0 potential here is infinite potential ok so what is the energy of electron or if it's a atom somehow I constrain the atom between two parallel plates right what's it allowable energy if you recall our two lectures ago we actually give a solution without solving the Schrodinger equation we say you have standing waves right so we have 1/2 wavelengths the full wavelength with that we found the energy before and what we're going to do next is using the Schrodinger q equation and to derive the same result so here energy is and here Y is 0 so for the 0 part next suppose I take my coordinate X is good between zero and D so between zero and D if you look high today Schrodinger equation is essentially the same equation right right because the potential is zero and then outside this region because protons are is the infinite so what's the wave function value this is infinite this has to be zero right so my boundary condition is essentially I this two point we function zero and in between my solution is here so I'm going to use this boundary condition use the solution we have so we have the solutions here I'm going to not to repeat that solution and in this case we have x equals zero wave function should be zero and that helps me here when x equals 0 this is 0 this is this is 1 this is 1 a plus B equals 1 so I have a plus B equals 1 that's the first of 0 sorry a plus B equals 0 thank you yep and X squares for D wave function 0 again so I have a a exponential minus ikx plus t i KD plus B X venture plus i KD equal 0 right so a equals mass B so if we'll a equals must be a substitute in this what I have I take the out I will have a negative to a I sign KD equal zero right after I substitute the first a equals minus B that's what I have okay so I cannot have a equals to 0 B equals zero if I have that my wave function 0 I have nothing there right so my only way to have this meaning for is sine K D equals 0 and that means a times D now has to be interviewed C right sine k D equals 0 plus R PI 2 pi right every to every PI so I have now I have to put a subscript n because this is the N times pi and the N could be 0 plus minus 1 plus minus 2 so those are the mathematical solutions okay and this mathematical solution tells me K n now is quantized right K n is n PI over D and the KN is 2m e n H bar square so 2m e n H bar square so my energy becomes discrete right so from here I solve for the energy of this particle eCourse uh-oh should I yeah square root 1/2 and I square that so it's a pie square n square and H bar Square and a 2m d squared so this is the energy solution for the particle in the in the old days it was it called Patagonian Box in that one dimensional box right and this is a comment before with the simplest quantizer solution and it was simply a homework right the first possible mathematical demonstration the simplest one and by say in the 70s people starting actually named on atomically flat film and to realize those simple solutions and now it becomes a very standard as you can you see from the last time we show from website a lot of people doing this quantum well using this energy quantization to make a lasers particular okay and if you substitute it because the last time we we use the H so if you substitute H equals H by cross H 2 PI over 2 pi you'll find that solution is identical to what we gave before so the last time I solve this problem we set the energy quantization without any real OC solving the equation and now we solve the equation we show that's the identical so this is the eigenvalue and of course with the eigenvalue we can also to go to say for each n there is also a corresponding Phi n the wave function right and it's important I carry this n number they quantize the the quantum number and so here Phi n you substitute back into the solution here I equals months B what we have the mass to I a and the sy n PI x over D okay and from this you will say and it should not be 0 if n is 0 again wavefunction is everywhere is 0 there's no particle there right and the symmetry also tells me what really is meaningful is N equals positive 1 2 3 so that's the LCD the quantum number N and how to determine this a the Lomo lies a ssin right so because this is a probability so the probability of finding the particle between 0 to D is why because that's the outside is 0 and inside must be somewhere so Milo Malaysian is 0 to D plus I and times simple sign and complex conjugate DX that should be equal to 1 I'm not going through the math you can find that a equals to i1 over to D and that means the wave function equals 2 over D sine M PI x over D okay so if you go to read a book ah the example given in the book take this wave function go to calculate what's the average momentum what's the uncertainty in the momentum and what's the position of the particle what's uncertainty of the position of the particle right because in the in the example which I'm not going through you will say from data you can actually show this the Heisenberg uncertainty principle is indeed the case of course it should be in those examples ok so you can take any of those n values and do that ok so ah what I want to emphasize this is a 1d problem one-dimensional problem a 1d problem I get a one quantum number N and what I'm going to Lex to show you is if I go to a two dimensional problem I'll get two quantum numbers okay so I'm doing a simpler simple extension of the same problem here is a film right and Lexia is a wire this is now people doing all nanotechnology from film to wire to dot so it's a 1d 2d and 3d 3d is a 0 D because when you constrain street dimension normally we have 3d right bulk material but if you constrain in three dimension becomes a zero D problem so let's do a 2d problem to the particle of quantum wire right so again I do a square and if you want to a cylinder you have to go to a cylindrical coordinate right I'm not going to spend much time on math I want to just tell you the concept so let's say this is a zero D I can do a square wire you can also do a rectangular wire whatever and outside is potential yo is again infinite inside potential yo is zero right so what's the energy of a particle now inside this wire right and we can go back and write down the Schrodinger equation steady-state Schrodinger equation outside this wire because the potential is infinite wave function is infinite now you zero we found in zero and inside U is 0 so I have 2d problem so I'm going to write down the 2d differential equation right so I have a partial differential equation again let me move this H bar square 2 m combined with the e to get a case where so I'm jumping one step I'm going to write the partial differential equation into the steady state part is so this is my x coordinate this is my y coordinate i'm equals K square plus I equal to 0 and this K square is I get is 2m e uy by H bar square that's the consequence I move this to here this is the equation inside this region and the boundary wave functions are 0 right so for this partial differential equation again you'll find that K include the energy energy of your lecturer eigen value of the solution process and because the two dimensional partial differential and the way I do it separation variable right so ah I do want to go through it to show you get to quantum numbers so this is the counting the quantum states what is the quantum state that concept so now I say ok my wave function is a function of x and y and I do separation of variables so I separated into a function X a function y and if I substitute it into the original differential equation I will get this is substitute the N X DX plus y the Y square plus K square equals zero right this is after you substitute the wave function into here you divide both sides by this wave function and you say because this is a function of X only so this is a function y only this is the K is the separation variable constant right so your only way is this is a constant this is a constant so that you add up because this is a function y this is a function of X this is a constant so this each of this term has to be a constant right so my location is I have DX square equals minus KX square so this is minus KX square mass KY square add up together that gives me K square okay so I have the first and if you look hi today my boundary condition without my boundary condition right at the a curve x equals 0 plus I equals 0 that also means this at 0 must be 0 because Y is any value plus right and same I say X square is P I have D equals 0 and you can write y equals 0 this same argument that will say y 0 is 0 why the is zero and this problem now if you check with this to boundary condition this equation is identical to the to what we had before I didn't write it's really you go to look at this solution with the same boundary condition your solution is the same so what we'll have based on this boundary condition I'll get a sigh K X times D equals 0 that's what we are written down for one D promise sign KD equals 0 in this one D now is the K in the X Direction 0 and similarly you can divide the ration why the ration apply the same you will get a sine KY times D equals 0 okay so those are the solution requirement that impose my KX KY okay so I'm going to find the day what do we have now is K X n that's my quantization number quantum number equals n pi divided by D and this is again a equals zero plus months 1 + KY I don't use the same n because the to the rations are independent right I use a different integer L let's say and that's L PI over D and this is l equals 0 plus minus 1 so after I substitute back into a KX square plus KY squared equals K square what I have K square is 2m e não é is a the quantum number include both NL h square - y equal KX square is the n pi over d square plus l pi over d square so now you can say I have two quantum number N and L because I have a to the ration two degrees of freedom and corresponding to this two quantum number i will also have poss-eye wave function l n equals a constant say Ln and sine n PI x over D sigh l pi y over D okay and again you can use the low molestation condition to find the coefficient that's not the important what is important is now to recognize that each set L and the N right when l equals 1 N equals 2 so when l equals 1 that's a half way in the x direction or another Ln I should do an L to be consistent okay so when N equals 1 that's a half way in x direction l equals 2 that's a full way right foot here in the y direction and now if I do N equals 2 that is a food purity in the X direction and L cross Y is a high purity in y direction so there are two different of forms right two different quantum state however the energy doesn't hear an equals 1 l equals 2 or English 2 l equals 1 right so this to quantum state have the same energy okay this is the degeneracy that Quantum's this energy level is degenerate so we have now the concept I say what is a quantum state right each wave function here is a quantum state and then say some quantum state a degenerate when they have different quantum numbers but same energy level okay and some of you are familiar with the density of states and this is the original density of states it's just so when we have many will contact more in the terms of 410 sub States rather than degenerating okay so ah these are they all these mathematics I'm going to use the rest are for the solution I'm going to give you right because they say you'll have more partial differential more involved solution of partial differential equations but also I'm going to show you my way of doing that without solving the equation I can do that pretty much for all the problems so I say that's my contribution to quantum mechanics because okay so let's look at some other problems we see this one you can say it's a partner in a box you can say it's a quantum well quantum work on dots or it's a model as an atom right it's a translational energy so they is they say kinetic energy pod is quantized and let's look at other examples we talked about before if I have a diatomic atom like a hydrogen atom a hydrogen molecule right diatomic molecule then you have a vibrational part of the energy you have rotational part of energy right so let's first look at the vibrational part if you have a mass spring system right the force in the harmonic Sprint picture we have will say force linear proportional to displacement right and we see the problem each problem is different it's because the potential so we need to find the potential and the potential energy is the derivative forces the derivative of the potato that's a always true right so from here I say the potential energy is one-half KX squared so this is the one that you should put it in the steady state the Schrodinger equation that's your potential now you go to solve that equation now you can see it's a complicated because my Patricia is a function of X my coefficient is no longer constant so solving this equation actually is a lot of effort you have to go see a lot more math than the one that we talked to before and let's see what they end the result right it's a one-dimensional problem right the variable is X only so from that I have one quantum number okay one degree of freedom one quantum number and the energy it turns out quantization one quantum number equals h nu n plus 1/2 and this new is not different from a classical oscillator if you have spring mass system what's the frequency square root of K over m and 2pi so you could of course write this h-bar Omega n plus 1/2 so you can see now the difference between a quantum and classical system is that say in a closure system you're just arbitrarily determined by the displacement amplitude of this business in the quantum the energy has to be multiple of this H nu right that's the vibration or energy quanta and this 1/2 is a zero-point energy okay zero-point energy and it turns out say this oscillator is also valid for the lattice when we have a lot of atoms in the solid and when we have electromagnetic wave wave it's also oscillator electromagnetic field oscillator all the energy is this form h-bar Omega n plus 1/2 and this 1/2 is a reflection of the uncertainty principle okay and it's very interesting this zero-point energy is also something that related to the 1 watch force and some people say even also relate to the missing mass of the universe okay and this there there there I can out comment more on this later so we have this is the harmonic oscillator and the next one is reach the ruler so what what I'm trying to do is is for example is two hydrogen atom for molecule right I see one mode of energy storage energy is this atom vibrator attitude to each other that's vibrational that's harmonic oscillator and the other way of storing the energy is the by rotation of the molecule right and so in that case I say reach the ruler so you just decompose the problem as fix the distance between the atom they will rotate and how many degrees of freedom here right the distance is fixed so R is fixed and they're only see the fight in space so I have two degrees of freedom right so for our rigid rotor going to have a mass m1 m2 and I fix the distance and the rotation is related to the moment of inertia which is the m1 m2 and R square R zero squared so that's a separation between the atoms m1 plus m2 okay and now in terms of degree freedom we function because it has two degree of freedom the wave function is L and this M is not mass when I use the subscript here that's an integer okay everybody use it so you have to distinguish yourself so LM are integers and of course we have to because FC defy two degrees of freedom right and I'm not going to write this is a spherical coordinate problem and that the solution is a pretty complicated by the later are your your CD I'll go to website and to show you it's much better than I draw on the board and so because of the wave function two quantum number I have also the energy but now the energy is degenerate so I'm not using the concept I just mention it turns out the for I have a several combination of L and M that can give you the same energy e l energy only depends on L wave function has two quantum numbers and so II L M and you see it doesn't depend on 2i l l plus one doesn't depend on M rotational energy level right and this quantum number l equals zero one two and that's a solution of the Schrodinger equation in fact that in this case is the angular momentum and M constraint is the absolute value is less than L that's equal to L okay so what does it have is if you have wave function 0 0 l equals 0 M is only 0 right now if wave function l equals 1 then M is mass 1 0 plus 1 right equals to 1 minus 1/2 sine 1 0 cosine 1 1 you say this 3 wave function have the same energy and to just give you a heads up what this way function look like is a dump I'll ship ok and so now we have the degeneracy of any energy level l quantum number is 2 L plus 1 because here M is 2 L plus 1 right for any L I give M is 12 plus 1 that's my degeneracy so now I'm counting the quantum state and the jit degeneracy for energy levels okay and this is the rotation we said ok first one is translation vibration rotation right if you think about Adam there's another one that's the electronic okay so let's go to the electronic energy level and the simplex electronic picture is a I have a hydrogen atom right 1c proton one electron so we have one nuclear we nuclear is much more heavier than electrons so you can be considered as stationary and now the electron will Abbott around the loop clean right so before we see the force to turn to charge what's the force Coulomb force force between two charged okay so Q 1 Q 2 Q 1 Q 2 4 PI epsilon that's the dielectric constant R square right so here is the separation and of course again if you integrate this you get the potential right the potential is the fourth derivative whether we read in there right so the potential for this problem U as a function of R is minus c1 R because the charge is opposite so that's an essay and then there's a negative side in the potential so when they bound very close they go to lower potential right so that's the direction they're given this potential if you look at this problem 1d or 3d problems electron moving around the loo clean its our it could be angular right so you have three degrees of freedom and my wave function is a function of R is function of theta a function of Phi spherical coding three dimensional and then I'm expecting three quantum numbers so I'm expecting three quantum number and L M okay so now I have the three quantum number and each set each combination plant number give me one quantum state but if you go to solve the problem for energy eigenvalue you find that depends endow me so although I have many quantum state and then my energy e n is a function so LM but it doesn't it doesn't depend on the LM so what I have is the mass of the electron C 1 square and the two H bar square else work and if you plug in the number you find this one is thirteen point six electron volt divided by n square so this is the our energy depends on n only see the wave function has three quantum number and what are the constraints for those quantum numbers here I have a integer 1 2 and L has to be l has to be less than n has to be positive so when N equals 1 l equals 0 okay M is the same constraint I have for this problem so ah M is equal or less than L okay so this is the energy level and if we look at a wave function n equals equals 1 and then I have l equals 0 1 0 0 right increase 1 and then when I have N equals 2 L could be 0 in that case M is 0 right or I can have wave function equals to L cross 1 because when N equals 2 I have 0 1 and then in that case I have 2 1 0 to 1 1 okay so the degeneracy GN for any let us say m value is n square n equals 1 is a one way function N equals 2 that's a full wave functions right but say N equals to I have 1 energy I have wood for wave function have the same energy ok so almost I'm almost here and Konkan but before I do that I want to say my way of solving the Schrodinger equation I didn't solve it right so now I'm going to show my hand waving way of solving I'll give you one example and you can do the others probably I said that in the previously when we study our quantum Wells the wave is either a half wave or full wave that's my boundary condition to form standing waves right so this one we already gone through and we said that's consistent just based on the standing wave requirement when we do the oscillator my potential is like this right 1/2 KX squared that's my potential we said that each problem what distinguish the problem of energy is real this potential and its corresponding boundary conditions right because the other terms look saved in the Schrodinger equation so now I look at my this is my X this is my potential yo I see what would be the wave function under this constraint the potential constraint again I'm going to make this hand waving standing wave requirement I don't even know I the boundary it has to be zero in fact it's not exactly zero so my solution will not be exactly correct but I'll just say now I still have my standing half half wave a full way right and this is at the boundary here this is my X N and now you say - xn is a half way must be motor those knots of the way with the right be the end point where I is that the boundary constraint has to are so here I have 2 X n right so that's the wave must be numb that tends to x n multiple of 1/2 wait that's my requirement and what does it it tells me it tells me X N equals 4 over n H bar H over X n no X H over P because lambda is H over P right I change the wave lengths into momentum okay and now I have with the from here I have the P so P in terms of X n I I will just argue that the kinetic energy of this particle in the oscillator equals potential energy okay so that's the what so the kinetic energy is a P Square to M equals potential energy 1/2 K x squared right KX n square P I have the here in terms of X N and from there I can find out from this I substitute the P into here I can find the X n so X n square equals M K and the NH over 4 and now I go because my this is my X n my en energy is the kinetic energy plus potential energy right I see there you go so the AEI equals KX n square ok so equals K X n square and you substitute the in and what I have is a 4 and H K over m and per my format is pretty close right if you look at the harmonic oscillator the real solution is en is H nu times n so the point is this quantum number is an in you're proposing here - yeah right you see there my potential because of a code potential constraint it changed my column quantization is a proportional to n and in the case of a quantum well this one you see is a proportional to n square right you go back to look at your year in proportion n square in this shape the yein changing to n and in the electronic potential that we just covered here here is 1 over r hn g or n into the denominator square so my point is all this problem are really defined by the potential of the particle that's the constraint okay and in fact I try the day electronic I can also make the same time with an argument I get a 1 over N squared I can get that 13.6 correct but I can get 1 over N square I haven't found out how I do this spherical problem gather L plus L times L plus 1 maybe someone can figure it out that's an angular momentum problem okay so ah now we have solution for translation we have vibration rotation electronic for atoms so we almost have everything right but how once then that's not even in classical mechanics and this is the spin it's not a you a in the Schrodinger equation you have to go to relativistic consideration to get that additional spin so I'm not going into spinning in detail it took me actually a long time to have some idea of spin complete quantum okay so I have legs is a spin right so when we solve a problem like a rigid rotor we the rotor is rotating rotating about the certain axis right that's an angular rotation that's the classical you have classical analogy right but see there is an intrinsic property of any particles mature waves there and that's an intrinsic see angular momentum and that angular momentum is characterized by the spin and turns out spin of electrons and photons photons they are different electron has a half spin and the photons photons have integral spin and because of this difference they obey very different rules okay so let's say electron spin and the quantum number for electrons being we use the s and there are only two one positive one half negative one 1/2 so those two quantum numbers you can get it you cannot get it from straw in Schrodinger equation so I'll just artificially added in to the wave function that we see for example when we consider adding we functioning from we'll add an S here okay that's the important for me to come and for our photons and photons and later on photons s is the integral integer one okay X equal one okay and it turns out for the half spin are particles this statistics is very different than the integer spin of particles and for this half spin particle we have Pauli exclusion principle so now if you go to say my quantum state for that electron right we just consider the electron problem the electron I have n L M and now I add s s equals plus 1/2 of minus 1/2 so I originally for NLM that the G N squared degeneracy now because of s I have to and square degeneration okay so are this one are what the Pauli exclusion principle say is each quantum state for this half spin each quantum state can have maximum one particle okay so that's the see it does not apply to fold on doesn't apply it to fold hung because they have in their spin but apply to electron each quantum stay you can only have maximum one so because of this fold Hong's electrons they behave very differently and this electrons is the fermion from Fermi and photons are bowls on from Bosse electron unlike each other and photons photons you can take as many as 1 1 count of state ok so this gives me for the case fo here right particle or SI electron you can see the degeneracy now is GN equals 2 and square it doesn't change the energy level because of the spin unless you go to very very detail but that's fine with just 2 N squared that's degeneration ok now I've gone through all this you see what's the use why are you going through this right so what do we what do we say starways is this lecture we want to understand at least appreciate the energy are quantized in the mature waves right electrons translational energy vibrational energy rotational energy electronic all those are quantized that they become discrete and we also emphasize that depend on the degree of freedom right we have quantum numbers and quantum state could be degenerate if their energy is the same but they are actually different quantum states they have the same energy levels okay so let's go to real work say where are we observe those quantum effects and if you look I say how the quantum mechanics started was actually from a blackbody radiation and also explaining the hydrogen spectrum right so observations okay and very clearly because the energy levels in the matter is discrete and that one interact with the other particles for example of photon interact with the molecular electronic energy level right and let's say the solution we have for the electronic energy level is negative thirteen point six and square electron volts right so N equals one that's one energy level n equals two is higher so let's say this is an increase 1 this is n equals 2 right an increase to and so when a photon whether photon interact or not depends on the photon energy matched to this energy level or not because the wind I think why electron in hydrogen the first sit in the lowest energy level all right and the photon energy if it comes to match so that it goes to the next level it gets absorbed right if it's impotent it doesn't get absorbed if it's too high doesn't get up though so what I'll say is you say discrete absorption lines right and in fact if you go to check the solar spectrum of sunlight going through the atom fear you see many discrete lines and that's because the atoms have discrete energy and they absorb full tongue of different energies okay so ah that's one example and what I want to do go next is go to the website and then let's look at this online and also periodical table that's fully merica absorption okay so let's start first with a solar spectrum maybe this is a good picture let's see how good it is okay here you can see ah this is a sunlight the I think the this is a blackbody from the equivalent and the yellow is outside the atmosphere right and once you go through the atmosphere it become very discrete and in this short wavelength so now you have to learn how to convert the wavelengths into energy and wavelength times frequency in equals speed of light right so if you know wavelengths you know frequency and frequency is each new energy energy if you convert further into electron volts and why electron volt is equivalent to 1 point 2 4 micron okay so that's the so here this is about one point two four that's where why you like homework shorter higher energy of photon longer lower energy right so here you go to this part it's electronic because the vibrational energy rotational energy don't have the high energy to interact with photon so it's a it's really the electrons in the molecule or atom absorb the photon once you go to here on this side it's a rotation and vibration rotation is actually a very small energy and our vibration is a higher energy so actually what what the molecule do for example it's when a vibrate also rotate so that's why actually it's a depend on your resolution your equipment if you do very high resolution you can actually say discrete lines of rotation vibration okay and there are many many lines and there are many species in the atmosphere they overlap with each other so you can see that's a it's the quantum mechanic doesn't have to be going into nano and it's actually everyday and let's go more first I want to show those are the solutions for our day wave function how the wave function psi and when you do precise precise complex conjugate what they really look like N equals 1 is a sphere right I think or two you can say it's like a dumbbell two hot blobs right and this is a when l equals zero it's a spherical when l equals 0 M has to be 0 right and 0 0 that's only so and then plus spin then you have two states let's being s if I think about electron going into N equals 1 then n equation one can have a two quantum state s equals plus minus 1/2 that it hasn't considered right so this could be two-speed two-stage N equals 1 N equals 2 is say if I add spin is 2 n square so equals 8 right equals two so each of this wave function sphere that's a l cross 0 now I only cost one that's a p orbital that's the atomic orbital T optho and when N equals 3 I have p orbital here and increase through agency and three zero zero that's an S orbital this spherical okay p orbital and then have D up to D orbital you can say it has a five wave functions five times two is ten right so there are ten a possible quantum states for the D orbital and our six here to here so what it is bringing you to this brings you to the periodic table now let's look at periodic tea right come to my office you should come to my office I look like that every day and the most of time we feel there should be more atoms there can't find it right so start from our hydrogen right one N equals one helium increase one there to spin up spin down okay so now let's look at the feeling right so N equals one I have hydrogen will be spin up or spin down English - is he new now the legs n is 13 point 6 divided by four so it's a higher a few electron volts higher right so now I have neom electron increase - now right perineum so now - n CS equals s s orbital field right and now I go to boron carbon so I have boron carbon the P orbitals 3p orbitals so that can hold the six more atoms so now you go to here go to helium so butan my why I say a bit Ania why helium is so stable because between here and here is a few electron volts right that's a large energy remember KT is 26 milli electron volts Millie right so it's not active because it takes a lot of energy for the charge to go to the next level so fully taken there okay so you continue so anchor 3 anchor 3 I should have say I have an inquiry 3 I have P I should have D right but since start to change because rather than D rather than if you the potassium rather than going here it started with the the next one they increase for what happens is to say well the solution I gave is a one charge one nuclear one charge right and that we have a lot of electrons they will interact that potential will change so the solution was splitted be another exactly they are not going exactly to be equal to each other there are small difference between those P and D levels and in fact that they are it turns out the 3d which are here this is 3d is less say energy then they say the higher energy then R 4 and S orbitals so the charge always goes to lows of the first right and so that's the day this is those transitional metals those are the transitional metals are the D orbital and once you go to here the F up do comes in ok very complicated once you go all those say wave function is complete wave functions if you do catalysis that this is a C is something the thermoelectrics people look into this all the time ok so I will stop here and the pretty much with this through lecture I went through the quantum mechanics you

Channel: MIT OpenCourseWare

Views: 40,658

Rating: 4.9437938 out of 5

Keywords: free particle, degeneracy, electron spin, photon, boson, electron, fermion, Pauli exclusion principle, translation, rotation, vibration, electronic potential, Coulomb force

Id: bESVLOTvijk

Length: 82min 12sec (4932 seconds)

Published: Wed Jan 16 2013