WEBVTT chemistry/physical-chemistry/hovasapian
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Hello and welcome back to www.educator.com and welcome back to Physical Chemistry.
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Today, we are going to continue our discussion of the postulates and principles of Quantum Mechanics.
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Postulate number 3, for a quantum mechanical system described by the wave function ψ,
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whenever a measurement is taken of the observable quantity associated with the operator A,
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the only values ever observed will be the Eigen values of A.
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That is the only values observed will be the A sub N that satisfy the equation A of ψ = A sub B of ψ.
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In other words, if I want to measure the kinetic energy of a particle, the kinetic energy operator.
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The only energies that I will ever going to find when I take the measurement, by observe, are going to be the Eigen values of the operator.
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It is very important postulate.
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Let us go ahead and work a little bit with it.
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Let us see, I will stick with black.
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Let us try to measure the kinetic energy of a particle in a 1 dimensional box.
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What are we going to find?
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Let us find out.
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Our kinetic energy operator is – H ̅/ 2 M D² DX².
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We are dealing with a 1 dimensional box, this is actually just a regular derivative.
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It is not a partial derivative.
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I hope you forgive me if I just keep that partial notation.
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Our ψ sub N for a 1 dimensional box is equal to 2/ A¹/2.
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You remember sin N π/ A × X, those are the wave functions for the particle in a 1 dimensional box.
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Let us see what we have got.
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Let us operate on it.
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The ψ sub N is equal to.
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That is fine, let us go ahead and write it out.
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H ̅/ 2 M and I'm operating on the function 2/ A¹/2.
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We want to write as much as possible and when you are doing these quantum mechanical problems and
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calculations because there are these symbols all over the place.
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You definitely want to write out as much as you can.
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You will lose your way, that is what everybody does, that is just the nature of the game.
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Sin N π/ A × X.
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I’m going to go ahead and differentiate.
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I’m going to take ψ.
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I’m not concentrating here so let me do this again.
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We have D² DX² and we are operating on the function 2/ A ^½ × the sin of N π/ AX.
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We are going to differentiate this function twice and then multiply by this thing.
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When I differentiate the first time, I’m going to get N π/ A × 2/ A¹/2 cos N π/ A × X.
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When I differentiate it the second time, I'm going to end up with N² π² / A² × 2/ A¹/2 and
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it is going to be negative because the derivative of cos is negative sin.
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It is going to be sin N π/ A × X.
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What I end up getting here is, this is going to be equaling - H/ 2 M × N² π²/ A² × 2/ A¹/2.
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This is actually a negative because that is negative.
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Sin N π/ A × X and what I end up with is, the negatives cancel, this is H ̅.
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I get H ̅ N² π²/ 2 MA² × 2/ A¹/2 × the sin of N π/ A × X.
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And that is equal to the operator.
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When I operated on it, what I end up with is.
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This thing right here, that is my ψ and that thing right there that is my A sub N, that is my Eigen value.
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In this particular case, this wave function it happens to be an Eigen function of the kinetic energy operator.
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In other words, when I operate on this function with a kinetic energy operator, I end up getting the function back × some constant.
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This constant is an Eigen value so when I get this, that means this is an Eigen function.
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Ψ sub N is an Eigen function of the kinetic energy operator.
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Any measurement that I make of the kinetic energy of the particle in a box is going to end up giving me,
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What I get is going to be the Eigen value.
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Any measurement I make of the kinetic energy of the system will give me H ̅ N² π²/ 2 MA² 2 / A ^½,
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I hope that I have not forgotten some exponents somewhere.
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Depending on what N is, depending on N.
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N is the state of the system.
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Depending on the state of the system whatever N happens to be, remember N can be 123456.
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Those are the quantum numbers for the particle in a 1 dimensional box.
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The measurement that I make is going to be different.
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But whatever measurement I do make, the only measurement that I see,
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the only thing I'm going to observe is going to be an Eigen value of that particular operator,
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Because the wave function itself happens to be an Eigen function of the operator.
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That is what that particular postulate that we just saw says.
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In this case, let us go to blue.
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In this case, our wave function for the system ψ happened to be an Eigen function of the operator that we chose,
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which is the kinetic energy operator, in this case.
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In this case, our wave function ψ happen to be an Eigen function of our operator of interest which was K.
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This does not have to be the case.
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This need not be the case.
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In other words, you can have a wave function for a particular system and then try to measure the linear momentum of that system.
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You are going to have to apply the linear momentum operator to the wave function.
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But when you do that, you are going to discover that the wave function for the system is not an Eigen function of that operator.
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The question becomes when you measure it, what value you are going to get?
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We have already seen that when it is an Eigen function that we are just going to get the Eigen values, that is what we are going to see.
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We are going to deal with the case where it is not necessarily the case.
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This may not be the case.
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For simplicity, we just want to deal with some simple functions so we actually see what is going on
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rather than complicating it where the function is going to end up getting in the way of our understanding.
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Let me go back to black here.
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Let us take our wave function ψ equal to the sin of KX and let us choose our linear momentum operator in the X direction.
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Let us choose that as our operator of interest.
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We have sin of KX and we have our operator so let us go ahead and write down what it is that we have.
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We have ψ is equal to sin of K × X and we have the linear momentum operator which happens to be – I H ̅ DDX that is the operator.
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We are going to operate on this function.
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Operating on the ψ that is the same as taking the –I H DDX of this sin K of X.
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This happens to be - I H ̅ K cos of KX.
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Operating on the function sin of KX gives me - IHK cos KX.
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Cos is not sin, you did not end up operating on this function and getting some constant × the function itself.
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You ended up with a different function.
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Clearly, the sin X function, our wave function here is not an Eigen function of the linear momentum operator.
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Precisely, because it ended up not being equal to some constant × the function itself.
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I will write this separately.
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I do not what to do this here.
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Let me go to the next page.
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Clearly, ψ which is equal to sin K of X is not an Eigen function of our linear momentum operator.
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The question is what value do we get when we measure the linear momentum of this particular wave function of this particular particle?
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What value do we observe when we measure the momentum of the system, that is the question.
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Let us see what we can do here.
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We are going to start playing around with some mathematics here.
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Here is how we deal with this.
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There is a function E ⁺IKX, E ⁺I θ.
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The Euler’s formula E ⁺IKX.
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The exponential function actually factors into the cos and sin function.
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This is cos of KX + I × the sin of KX.
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There is also another function E ^- IKX that is equal to cos KX - I × the sin of KX.
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These are just basic mathematical relations that exist.
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Here is what is interesting.
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The wave function that we have, our wave function sin of KX, it can be written as a linear combination of E ⁺IKX and E ^- IKX.
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In other words, I can take E ⁺IKX which is this thing and E ⁻IKX which is this thing and
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I can end up writing our original wave function which a sin of KX as some combination of these two, multiplied by some constants.
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It is going to turn out that these two functions are Eigen functions of the linear momentum operator
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and because that is the case, I can extract some information.
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The wave function of KX can be written as a linear combination of E ⁺IKX and E ⁻IKX.
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Let us go ahead and see how.
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By the way, a linear combination of these two functions, a linear combination means some constant ×
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the first function + some constant × the second function.
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You might have 4 functions, a linear combination of 5 functions.
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It is just going to be some constant × that function + another constant × that function + another constant × that function,
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That is what linear combination means.
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You are combining them linearly, adding them up.
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Let us go ahead and see how it is represented.
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Our sin of KX, I can actually write it as 1/ 2 I × E ⁺IKX - 1/ 2 I × E ^- IKX.
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In this particular case, my coefficients are 1/2 I and -1 / 2 I, that is my C1 and C2.
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This sin of KX can be written this way.
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Let us go ahead and show that that is the case.
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Let us expand it.
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This is equal to 1/ 2 I E ⁺IKX.
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E ⁺IKX is equal to this thing.
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I’m going to write it all out, multiply out.
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1/2 I × the cos of KX + 1/2 I × I sin of KX.
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Let me put it over here.
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-1/ 2 I × this which is this so we get 1/2 I × the cos of KX - 1/2 I × I × the sin of KX.
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This and this, they cancel, the I's cancel.
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I'm left with ½ KX -- + ½ sin KX, I end up with sin KX which is my wave function.
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The wave function ψ equals the sin of KX can be written as a linear combination of E ⁺IKX and E ⁻IKX.
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And we just saw that it was.
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We said that these functions happen to be Eigen functions of the linear momentum operator.
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We have taken a function that is not an Eigen function of the linear momentum operator but
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we have expressed it as a linear combination of functions that are Eigen functions of the linear momentum operator.
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Let us go ahead and prove that that is the case first, before we continue.
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The linear momentum operator operating on E ⁺IKX is equal to – I H ̅ DDX of E ⁺IKX is equal to - I H ̅ IK E ⁺IKX which is equal to,
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I × I is -1, - -1 ends up being H ̅ KE ⁺IKX.
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Clearly, linear momentum operator operating on IKX equals some constant × IKX.
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E ⁺IKX is an Eigen function of the linear momentum operator and the same thing the other way around.
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The same thing with when I apply this to E ^- IKX.
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I’m just demonstrating that these are Eigen functions.
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- I H ̅ DDX of E ^- IKX and what we end up with is - I H ̅ × - I K × E ^- IKX that equals.
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I and I is -1.
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-1 is negative so we end up with –H ̅ KE ^- IKX.
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Clearly, the linear momentum operator operating on this function gives me some constant × that function.
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E ⁺IKX and E ⁻IKX are Eigen functions of the linear momentum operator.
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Even though, ψ = KX is not an Eigen function of the linear momentum operator,
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it can be written as a linear combination of functions that are Eigen functions.
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When we measure the linear momentum, sometimes we get H ̅ K, one of the Eigen values, the Eigen value,
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one of the functions that is an Eigen function.
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Sometimes, we get – HK.
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HK and -HK happen to be the Eigen values of the functions that are Eigen functions of the linear momentum operator.
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That is all that we have done.
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I hope that makes sense.
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Let us see what we have got next.
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Let us go ahead.
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Postulate 4, this is what we are leading to this particular discussion.
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When a measurement is being made on an observable quantity corresponding to an operator and
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the wave function of the system is expressed as a linear combination, also called the super position of Eigen functions of the operator like what we just did.
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If the wave function itself is not an Eigen function of the operator but
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it can be expressed as a linear combination of functions that are Eigen functions of the operator,
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where the ψ sub I or expansion coefficients and they can be complex,
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The F sub I or the Eigen functions of the operator and each measurement will give an Eigen value of the operator.
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This is the definition of Eigen function Eigen value.
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That is the Eigen function of the operator, that is the Eigen value.
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When you end up taking the operator operating on it and getting some constant × that function itself.
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In the previous example, we had HK and – HK.
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There were two of them.
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The coefficients were the same, 1/ 2I 1/ 2 I.
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Because the coefficients are the same, when you take the measurement,
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half the time you are going to end up with HK and the other half of the measurements you get are going to be – HK.
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In general, if you have more than two functions and if these coefficients are not necessarily equal,
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there is going to be a distribution of how many times you get an Eigen value.
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The probability of a given Eigen value appearing as a measure of value is going to be proportional to square of the modulus of the actual coefficient.
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If the wave function is normalized, then the probability is exactly that squared.
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This is an expression of what it is that we have actually done.
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Let me say it again.
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When a measurement is being made of observable quantity corresponding to an operator and the wave function ψ
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is expressed as a linear combinations of Eigen functions of the operator or the ψ sub I
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are the coefficients and the N sub I are the Eigen functions of the operator.
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Each measurement will give an Eigen value of the operator A corresponding to the particular function.
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The probability that a given Eigen value appearing as a measured value is going to be proportional
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to the square of the coefficient, or the square of the model.
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I just say the coefficient because in case it is a complex number, it is going to be the modulus, the length.
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If the wave function happens to be already normalized, then the probability is exactly the square of the modulus of the coefficient.
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Let us go to our next postulate.
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When we take a bunch of individual measurements, we are going to get this value.
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What if we want an average value?
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Let us say we end up taking 1000 measurements?
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There is going to be some average especially if the values are not the same over and over again.
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If they have different values, there is going to be some average.
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Again, sometimes, like we said the probabilities are not necessarily uniform or equal, it is not going to be half and half.
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Sometimes, it will be 75% of the time you are going to get one value and 25% of the time you are going to get another Eigen value.
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What is the average going to be when we want to find the average value of many measurements?
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If ψ is a normalize wave function of a quantum mechanical system in a given state and the average value
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also called an expectation value of the observable quantity associated with the operator is given by this thing right here.
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In other words, what we do is we take the function.
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We operate on a function and then we multiply.
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What are those that we have got when we have operated on it by the complex conjugate of the function itself and then
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we integrate that function over our region of interest because that is going to give us a numerical value of the average.
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We already saw that the wave function for a system need not be an Eigen function of the operator of interest.
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We already saw that with our sin KX.
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Something interesting happens with the average values.
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When that the ψ, the wave function of a given quantum mechanical system is the Eigen value of the specified operator.
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We began this lesson by taking a look at what individual value we are going to get when we take one measurement.
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We are talking about an average value.
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Let us say I do not want to worry about individual measurement, I just want to calculate an average value if I were to take 10,000 measurements.
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This integral will allow me to do that.
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Something interesting happens when the average values, when the actual wave function itself is an Eigen function of the operator.
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It is exactly what you think.
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It is what we already saw in the beginning.
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We are going to end up with one value over and over again.
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We are going to confirm that mathematically.
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Ψ is the wave function of a given system and we said that the ψ also satisfies, is an Eigen function of a particular operator.
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It also satisfies the following relation.
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A of ψ = λ × ψ.
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Then, our postulate 5, the average value of A we said was equal this, the ψ conjugate × having operated on ψ, this is going to equal to,
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A of ψ is this so I’m going to put that in here.
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I’m going to get the ψ conjugate × the λ of ψ.
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This is now just a constant, it is our scalar so I can pull it out.
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It equals λ × the integral of ψ* and ψ.
00:30:40.600 --> 00:30:43.800
Wave function is normalized so this is the normalization condition.
00:30:43.800 --> 00:30:47.400
This is just λ × 1.
00:30:47.400 --> 00:30:48.400
I end up with λ.
00:30:48.400 --> 00:30:56.500
The average value is the actual Eigen value itself.
00:30:56.500 --> 00:31:10.000
Furthermore, this is saying that the average value, if I take 10,000 measurements I’m going to end up with the Eigen value.
00:31:10.000 --> 00:31:14.400
Earlier in this lesson, we talked about the individual measurements that we make.
00:31:14.400 --> 00:31:22.500
An average value, if I take if I take the measurement and if I get 1, 10, 1, 10, 1, 10, 1, 10, 1, 10.
00:31:22.500 --> 00:31:24.600
The average value is going to be 5.
00:31:24.600 --> 00:31:30.600
If I do another experiment and I get 5555555, the average value is going to be 5.
00:31:30.600 --> 00:31:32.700
What is the difference between those two situations?
00:31:32.700 --> 00:31:46.700
The average value is the same but in one of them, I only got 5 and the other one I got 1, 10, 1, 10, 1, 10 or I could have gotten 110, 29, 38, 47,
00:31:46.700 --> 00:31:49.400
I still end up with an average of 5.
00:31:49.400 --> 00:31:53.100
This alone does not really tell me anything just yet.
00:31:53.100 --> 00:31:55.300
This is gives me an average value.
00:31:55.300 --> 00:32:01.000
What I need to find is I need to find out the actual variance, the standard deviation and
00:32:01.000 --> 00:32:08.000
if that equals 0 then that means that this λ, that 5 is going to be the only value that I always get.
00:32:08.000 --> 00:32:10.700
That is what is going on here.
00:32:10.700 --> 00:32:14.200
Let us go ahead and calculate some other things.
00:32:14.200 --> 00:32:18.000
Let us calculate the second moment A sub².
00:32:18.000 --> 00:32:26.600
That is equal to ψ sub * A² of ψ.
00:32:26.600 --> 00:32:32.600
We know that the square of an operator is just the operator applied twice.
00:32:32.600 --> 00:32:50.200
It is going to be ψ sub * A, applied to A of ψ which equals the integral of ψ sub * A hat.
00:32:50.200 --> 00:32:55.300
This is going to be λ × ψ.
00:32:55.300 --> 00:33:11.500
The λ comes out so we end up with λ × the integral of ψ sub * A of ψ which equals λ × the integral of ψ*,
00:33:11.500 --> 00:33:14.000
This is λ ψ.
00:33:14.000 --> 00:33:20.500
This λ also comes out and ends up being λ² × the integral of that.
00:33:20.500 --> 00:33:25.900
This is just 1 by the normalization condition because the wave function is normalized.
00:33:25.900 --> 00:33:29.400
We end up with the λ².
00:33:29.400 --> 00:33:34.500
Our variance of this particular operators, particular measurement.
00:33:34.500 --> 00:33:51.900
The variance by definition is the average value of the second moment - the square of the average value, that is equal to λ² - λ² which equals 0,
00:33:51.900 --> 00:33:54.800
Which means that our standard deviation is equal to 0.
00:33:54.800 --> 00:33:58.900
The standard deviation is the square root of the variance.
00:33:58.900 --> 00:34:02.700
A standard deviation of 0, it is a measure.
00:34:02.700 --> 00:34:09.400
Remember what a standard deviation was, it is a measure of how far in general any individual measurement is from the mean.
00:34:09.400 --> 00:34:14.100
The fact that the standard deviation is 0 tells you that every measurement is one number.
00:34:14.100 --> 00:34:21.100
You keep getting λ over and over again, when you take, when you operate.
00:34:21.100 --> 00:34:34.400
When the wave function itself is an Eigen function of the operator of interest, for an average value you end up just getting the Eigen value.
00:34:34.400 --> 00:34:41.900
But more than that, it says that is the only value you get every time you take the measurement over and over again.
00:34:41.900 --> 00:34:45.500
This standard deviation of 0 says there are no other values.
00:34:45.500 --> 00:34:50.900
There is no average, you are just getting 5555555 over and over again.
00:34:50.900 --> 00:34:55.800
It is just 5.
00:34:55.800 --> 00:35:13.600
Let me go ahead and write that down.
00:35:13.600 --> 00:35:20.700
Let me go back to blue and write this.
00:35:20.700 --> 00:35:47.200
A variance equal to 0 or a standard deviation equal to 0 means every measurement of an observable,
00:35:47.200 --> 00:35:50.400
whatever the observable happens to be, whatever operator,
00:35:50.400 --> 00:36:28.000
every measurement of an observable associated with a generic operator A will give only λ, the Eigen value of A.
00:36:28.000 --> 00:36:32.600
In other words, there is no average per say of multiple Eigen values.
00:36:32.600 --> 00:36:36.900
It is the average of the same number over and over again.
00:36:36.900 --> 00:36:42.300
Let us go ahead and finish off.
00:36:42.300 --> 00:36:53.700
For systems whose wave functions are not Eigen functions of a given operator that can be expressed as
00:36:53.700 --> 00:37:02.600
a linear combination of Eigen functions of the operator which is this thing, the average value is this thing.
00:37:02.600 --> 00:37:09.900
Basically, all I do if I want to calculate an average value, I take the coefficients,
00:37:09.900 --> 00:37:16.400
I find the modulus, I square them, and I multiply it by the particular Eigen value of that function.
00:37:16.400 --> 00:37:21.000
And then add to it the square of the next coefficient × it Eigen value.
00:37:21.000 --> 00:37:24.300
The square of the next coefficient × its Eigen value.
00:37:24.300 --> 00:37:28.100
That will give me the average value.
00:37:28.100 --> 00:37:34.300
This shows that the average value of an observable is a weighted average of the Eigen values of the operator,
00:37:34.300 --> 00:37:41.100
Where the weights are equal to the square of the modulus of the expansion coefficients.
00:37:41.100 --> 00:37:47.600
If I have some function which is not an Eigen function of my operator of interest and
00:37:47.600 --> 00:37:58.600
let us say 75% of the time I end up with 2 and 25% of the time I end up with 5.
00:37:58.600 --> 00:38:07.300
My average value is going to be the coefficient squared of the first function × 2.
00:38:07.300 --> 00:38:11.400
The coefficients squared of the second function × 5.
00:38:11.400 --> 00:38:13.800
That is going to end up giving me some average value.
00:38:13.800 --> 00:38:16.200
It is going to be between 2 and 5 somewhere.
00:38:16.200 --> 00:38:18.900
That is what this is saying.
00:38:18.900 --> 00:38:24.100
This postulates that we have discussed in this lesson, they have to do with
00:38:24.100 --> 00:38:28.700
what the individual values are going to be when I take individual measurements.
00:38:28.700 --> 00:38:34.400
And what the average values are going to be when I want to calculate an average value and
00:38:34.400 --> 00:38:40.900
see what it is that I expect to get in general, when I take 1000, 10000, 10,000,000 measurements.
00:38:40.900 --> 00:38:44.900
That is what is going on here.
00:38:44.900 --> 00:38:50.600
I do apologize that these individual lessons have not really included a lot of the example problems.
00:38:50.600 --> 00:38:58.200
Again, I want to present the theory in a continuous way.
00:38:58.200 --> 00:39:02.300
I want to go ahead and do the example problems in bulk, because when I do that,
00:39:02.300 --> 00:39:09.400
it is going to allow us a chance to actually review the material rather than presenting some concept, doing example,
00:39:09.400 --> 00:39:12.000
presenting some concept, doing an example.
00:39:12.000 --> 00:39:18.300
If some of this is not exactly entirely clear, do not worry about it because we are going to be revisiting all of these and
00:39:18.300 --> 00:39:23.400
writing things over and over again when we do the examples.
00:39:23.400 --> 00:39:25.200
Terrific, we will go ahead and stop this lesson here.
00:39:25.200 --> 00:39:27.700
Thank you so much for joining us here at www.educator.com.
00:39:27.700 --> 00:39:28.000
We will see you next time, bye.