Normalization in Quantum Physics

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In quantum physics, a physical system such as a particle or a collection of particles is denoted by the ketsai which can mathematically be represented by a column vector. Find out about normalization in quantum physics with help from an applied physics professional in this free video clip.

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Video Transcript

Hello, my name is Walter Unglaub, and this is normalization and quantum physics. So in quantum physics, a physical system such as a particle or a collection of particles is denoted by the ketsai which can mathematically be represented by a column vector. Now for this particular simple example I'm going to consider a box that has sides of infinite potential. And a particle such as an electron that is trapped inside of this box. The length of the box is L and this is a one dimensional box so the particle can have a ground state wave function or a higher energy wave function but regardless of the energy of the particle, the wave function must be equal to zero at the boundaries. Why? Because the probability of finding the particle is going to be equal to the modulous squared of the inner product of that particle state with itself. And one of the postulates in quantum mechanics and in physics is that total probability must be conserved. And that the total probability has to be equal to one. Meaning that when you make a measurement you have to find the particle somewhere inside of this box. For this particular scenario since the particle cannot tunnel or leave otherwise this system. In position representation the state of this particle can be given for the wave function generally as A sign knx where A is an amplitude for the wave function and kn is the wave number for the particle. Kn is given as pie over the length L. Where n is an integer corresponding to the energy of the particle. So the only thing we don't know is this amplitude A. But we do have a constraint which is that the total probability must be equal to one. So in this case if we add up all the probabilities of the particle being in all of the positions we can determine what this value A is and we would do that by integrating the space dependent wave function over all space. So we end up with a probability density function that we integrate over set that equal to one and solve for the amplitude. So in this case we have one is equal to the integral over all space which in this case is zero to length L of the complex conjugate of my space dependent wave function times the wave function and it's integrated over x. When I plug in this analytic function for sai and I perform the integral I end up with one is equal to A squared times L over two. Therefore when we solve for A we see that A is equal to the square root of two over the length L. So the point is the larger the box or cavity in which my quantum particle exists, the smaller the amplitude for all the different wave functions that the particle can have depending on its energy. And now when I plug in this value for A my wave function is now normalized. So when I sum up all the probabilities I get a total probability equal to one. My name is Walter Unglaub and this is normalization in quantum physics.


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