Lecture 8 Symmetries, conserved quantities, and the labeling of states Angular Momentum

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1 Lecture 8 Syetries, conserved quantities, and the labeling of states Angular Moentu Today s Progra: 1. Syetries and conserved quantities labeling of states. hrenfest Theore the greatest theore of all ties (in Prof. Anikeeva s opinion) 3. Angular oentu in QM 4. Finding the eigenfunctions of Lˆ and Lˆz - the spherical haronics. Questions you will by able to answer by the end of today s lecture 1. How are constants of otion used to label states?. How to use hrenfest theore to deterine whether the physical quantity is a constant of otion (i.e. does not change in tie)? 3. What is the connection between syetries and constant of otion? 4. What are the properties of conservative systes? 5. What is the dispersion relation for a free particle, why are and k used as labels? 6. How to derive the orbital angular oentu observable? 7. How to check if a given vector observable is in fact an angular oentu? References: 1. Introductory Quantu and Statistical Mechanics, Hagelstein, Senturia, Orlando.. Principles of Quantu Mechanics, Shankar

2 Syetries, conserved quantities and constants of otion how do we identify and label states (good quantu nubers) The connection between syetries and conserved quantities: In the previous section we showed that the Hailtonian function plays a ajor role in our understanding of quantu echanics using it we could find both the eigenfunctions of the Hailtonian and the tie evolution of the syste. What do we ean by when we say an object is syetric?what we ean is that if we take the object perfor a particular operation on it and then copare the result to the initial situation they are indistinguishable.when one speaks of a syetry it is critical to state syetric with respect to which operation. How do syetries anifest theselves in equations?let us suppose that your syste is syetric with respect to translations in x that would iply that any physical property could not have an x dependence. In particular the energy would not have an explicit dependence on x thus: dp H x, p 0 p const x dt The oentu in this case is called a constant of otion. This illustrates a fundaental connection between syetries and conserved quantities. In fact, every syetry in a physical syste iplies an associated conserved quantity. In quantu echanics the tie evolution of an observable is describe by the following equation: hrenfest Theore: d dt  1 i Â, Ĥ  t d dt x  x 1 i x Â, Ĥ x x  t x Consequently in order for a physical quantity to be a constant of otion the corresponding observable has to obey the following relationships:  0 d t A ˆ 0 Â, Ĥ dt 0 States are labeled by specific values of their properties, which do not change with tie these properties are called constants of otion. We learned that in QM physical properties are represented by operators and that the values of properties obtained in easureents are eigenvalues of the corresponding operators. Hence the eigenvalues are used as the labels.

3 Conserved Quantities xaple 0: Conservative Systes The siplest exaple of a conserved quantity is nergy. Two lectures ago we have considered systes, where Hailtonian does not depend on tie explicitly. Obviously Hailtonian coutes with itself and consequently energy is conserved and can be used as a label for a state. Ĥ 0 t d Ĥ, Ĥ dt 0 Reinder: Schrodinger s equation: V Ĥ 0 const u x x,t r r,t r,t t i This type of differential equation is separable, i.e. we can look for a solution in the following for: r, t r t. Let s substitute it into the Schrodinger s equation above: V r r t i t r t 1 r 1 V r i r t t t u x e i t r and the right side of the Note that the left side of the equation only depends on position equation only depends on tie t. This can only be true when both sides of the equation are constant for energy. Then the equation above splits into the following two equations: d I i t dt II V t t e i r r r u, x Then the solutions to tie-dependent Schrodinger s equation will have a for: r, t r u t u r e i t In general, since the Hailtonian ay have any eigenvalues and corresponding eigenfunctions, the solution for this syste is a linear cobination of all the possible solutions corresponding to different energies: r, t C u r e i t, where C are the coefficients that can be deterined fro the initial and boundary conditions. This is a very iportant result: If we know the special wavefunctions (Hailtonian eigenfunctions) we can easily find tie evolution of this conserveative syste. If we know and u r then we know r, t at any tie! 3

4 Conserved Quantities xaple I: Particle in free space Labeling of states is particularly iportant when the energy eigenvalues are degenerate such as in the case of the particle in free space: Ĥ x x x x u x i e x i e Using the energy as a label doesn t copletely and uniquely specify a state. What about oentu? If oentu is a constant of otion then we can use it as an additional label to uniquely specify the eigenstate. We have shown earlier that the oentu operator coutes with free space Hailtonian, and since it does not explicitly depend on tie then oentu is indeed a constant of otion. pˆ pˆ i 0 x t d pˆ 0 dt pˆ, Ĥ 0 Therefore the oentu is a conserved quantity and its eigenvalues can be used to label the states. Then the unique labels for the eigenfunctions above would be: x x e ikx u,k x, u,k x e ikx k In fact we represent all the eigenfunctions (eigenstates) of the free space Hailtonian and the oentu on the vs k plot. very point on this plot uniquely and copletely specifies the state. 4

5 Conserved Quantities xaple II: Parity operator and syetric potentials Definition of a parity operator: ˆ x x What are the eigenfunctions and eigenvalues of the parity operator: ˆ ux ux ˆ u ˆ x ˆ ux ˆ u ˆ x u ˆ x ux ux 1 The eigenfunctions of the parity operator all are either odd or even. f ( x)= f (x) even f ( x) = f (x) odd Does Hailtonian for Siple Haronic Oscillator coute with the parity operator? x x x H ˆ x x Ĥ x x x H ˆ x Let s check the coutator: ˆ 1 1, Ĥ x Ĥ x x x x x x x x ˆ Ĥ ˆ ˆ 1 1 x x x x 0 x x This eans that one can always find a set of eigenfunctions coon to Ĥ and ˆ. In fact, last lecture we have shown that SHO eigenfunctions are always even or odd. 5

6 Conserved Quantities xaple II: Angular oentu and spherical syetry Consider a syste with a spherical syetry, such as Hydrogen ato: Hydrogen ato consists of a proton with a positive charge q = e = C an electron charge q = e. Consequently they are bound by a Coulobic potential: q 1 Vr e 40 r r Where: ε 0 = F/ is the perittivity of free space. As you can notice this potential does not depend on the respective position of the proton and the electron but only on the distance between the, i.e. if electron revolves by a certain angle around the proton the potential will stay the sae. In the syste with such spherical syetry, i.e. independence of the Hailtonian on the direction of the position vector just the length, the angular oentu is conserved this is very iportant as we will use this conservation later to use it for labeling of the states associated with a spherically syetric syste. But first let s just find out what is angular oentu in quantu echanics, as usual we use the classical echanics to build soe intuition. Classical angular oentu is a vector quantity defined as: L r p î î î x y z x y z î yp zp î zp xp î xp yp î L î L î L p p p x y z x z y y x z z y x x x y y z z In quantu echanics we can define a corresponding observable (Heritian operator): î x î y î z Lˆ rˆ pˆ x y z î x iy iz î y iz ix î z ix iy z y x z y x i i i x y z Lˆ î xlˆ x î ylˆ y î zlˆ z While Descartes s coordinate syste allows us to intuitively derive the coponents of the angular oentu operator, the spherical syetry of the proble akes it ore logical to ove to the spherical coordinate syste: x rsin cos y rsin sin z r cos 6

7 In spherical coordinates we can rewrite all the coponents of the angular oentu: Lˆ ŷpˆ ẑˆ cos tan x z p y isin sin Lˆ y ẑpˆ x xˆpˆ z icos tan Lˆ z xˆpˆ y ŷpˆ x i 1 1 Lˆ tan sin We call this operator the orbital angular oentu since it has a classical equivalent. If one looks at the coutation relations of this operator one finds: Lˆ Lˆ x, y ilˆ z Lˆ Lˆ y, z ilˆ x Lˆ, Lˆ ilˆ z x y Note: In QM there is another very iportant quantity, which is called spin. It turns out that this quantity also obeys the sae coutation rules and is therefore also called an spin angular oentu. The spin angular oentu has no classical equivalent. One can show that: Lˆ, Lˆ z 0 Therefore we now that one can fin d a set of eigenfunctions coon to Lˆ and Lˆ z : Lˆ Y z l, Y l, LˆY l, ll 1Y l, Where: Y l l 1 l! 1 i Pl cos e, 0 4 l!, l 1 l! i P l cos e, 0 4 l! Since the wave function needs to be continuous at, is an integer. One can show that l also needs to be an integer and that l l. l 0. P l cos are Legendre polynoials and the eigenfunctions Y l, are called spherical haronics: 7

8 Below are the plots of first several spherical haronics: Weisstein, ric W. "Spherical Haronic." Fro MathWorld - A Wolfra Web Resource. Used with perission. 8

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