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2D square box – separation of variables n y =1,2,3,….n x =1,2,3,…. Ground state of 1D infinite square well. n y =1,2,3,…. n x =1,2,3,….

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2D square box n y =1,2,3,….n x =1,2,3,…. What is the energy of the ground state of the 2D infinite square well? A)0 B)E 0 C)2E 0 D)4E 0 E)5E 0 n x =1, n y =1 E=E 0 (1 2 +1 2 )

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2D square box n y =1,2,3,….n x =1,2,3,…. What is the energy of the first excited state of the 2D infinite sqaure well? A)2E 0 B)3E 0 C)4E 0 D)5E 0 E)8E0 n x =1, n y =2 E=E 0 (1 2 +2 2 ) Or n x =1, n y =2

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2D square box – probability densities n y =1,2,3,….n x =1,2,3,…. Contour maps of | (x,y)| 2 Ground state (1,1)

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Q. The potential seen by the electron in a hydrogen atom is… A.Independent of distance B.Spherically symmetric C.An example of a central force potential D.Constant E.More than one of the above The potential seen by the electron is Spherically symmetric (doesn’t depend on direction). It depends only on distance from proton so it is a central force potential. On your own – no discussion

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3-D central force problems The hydrogen atom is an example of a 3D central force problem. The potential energy depends only on the distance from a point (spherically symmetric) x y z r Spherical coordinates is the natural coordinate system for this problem. What is the x-coordinate of our point (r, )? A) B) C)

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3-D central force problems The hydrogen atom is an example of a 3D central force problem. The potential energy depends only on the distance from a point (spherically symmetric) x y z r Spherical coordinates is the natural coordinate system for this problem. Engineering & math types sometimes swap and . General potential: V(r, , ) Central force potential: V(r) The Time Independent Schrödinger Equation (TISE) becomes: We can use separation of variables so

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3-D central force problems The hydrogen atom is an example of a 3D central force problem. The potential energy depends only on the distance from a point (spherically symmetric) x y z r Spherical coordinates is the natural coordinate system for this problem. Engineering & math types sometimes swap and . General potential: V(r, , ) Central force potential: V(r) The Time Independent Schrödinger Equation (TISE) becomes: We can use separation of variables so

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The part The variable only appears in the TISE as So we should not be surprised that the solution is Note that m is a separation variable and not the electron mass. We use m e for the electron mass. x y z r Are there any constraints on m? What can we say about and ? They have to be the same! Since cosine and sine have periods of , as long as m is an integer (positive, negative, or 0) the constraint is satisfied.

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Angular momentum about the z-axis is quantized: Note is similar to which is the solution to the free particle with Angular momentum quantization about z-axis x y z r As k gives the momentum in the x direction, m gives the momentum in the direction (angular momentum). There is nothing truly special about the z-axis. We can point the z-axis anywhere we want to. It is just the nature of the coordinate system that treats the z-axis different than the x and y axes.

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The part x y z r The solution to the part is more complicated so we skip the details of the solution. The end result is that there is another quantum variable ℓ which must be a non-negative integer and ℓ ≥ |m|. The ℓ variable quantizes the total angular momentum: Note that L Z cannot be larger than the total L.

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Total angular momentum is ℓ can be 0, 1, 2, 3, … The z-component of the angular momentum is where m can be 0, ±1, ±2, … ±ℓ

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