Orbital Angular Momentum, Hydrogen Atom (Part - 1) - Angular Momentum, CSIR-NET Physical Sciences Physics Notes | EduRev

Physics for IIT JAM, UGC - NET, CSIR NET

Created by: Akhilesh Thakur

Physics : Orbital Angular Momentum, Hydrogen Atom (Part - 1) - Angular Momentum, CSIR-NET Physical Sciences Physics Notes | EduRev

The document Orbital Angular Momentum, Hydrogen Atom (Part - 1) - Angular Momentum, CSIR-NET Physical Sciences Physics Notes | EduRev is a part of the Physics Course Physics for IIT JAM, UGC - NET, CSIR NET.
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Introduction

Angular momentum plays a central role in both classical and quantum mechanics. In classical mechanics, all isolated systems conserve angular momentum (as well as energy and linear momentum); this fact reduces considerably the amount of work required in calculating tra jectories of planets, rotation of rigid bodies, and many more.

Similarly, in quantum mechanics, angular momentum plays a central role in understanding the structure of atoms, as well as other quantum problems that involve rotational symmetry.

Like other observable quantities, angular momentum is described in QM by an operator. This is in fact a vector operator, similar to momentum operator. However, as we will shortly see, contrary to the linear momentum operator, the three components of the angular momentum operator do not commute.
In QM, there are several angular momentum operators: the total angular momentum (usually denoted by Orbital Angular Momentum, Hydrogen Atom (Part - 1) - Angular Momentum, CSIR-NET Physical Sciences Physics Notes | EduRev, the orbital angular momentum (usually denoted by Orbital Angular Momentum, Hydrogen Atom (Part - 1) - Angular Momentum, CSIR-NET Physical Sciences Physics Notes | EduRev ) and the intrinsic, or spin angular momentum (denoted by Orbital Angular Momentum, Hydrogen Atom (Part - 1) - Angular Momentum, CSIR-NET Physical Sciences Physics Notes | EduRev. This last one (spin) has no classical analogue. Confusingly, the term “angular momentum” can refer to either the total angular momentum, or to the orbital angular momentum.
The classical definition of the orbital angular momentum, Orbital Angular Momentum, Hydrogen Atom (Part - 1) - Angular Momentum, CSIR-NET Physical Sciences Physics Notes | EduRev can be carried directly to QM by reinterpreting Orbital Angular Momentum, Hydrogen Atom (Part - 1) - Angular Momentum, CSIR-NET Physical Sciences Physics Notes | EduRev as the operators associated with the position and the linear momentum.

The spin operator, S, represents another type of angular momentum, associated with “intrinsic rotation” of a particle around an axis; Spin is an intrinsic property of a particle (nearly all elementary particles have spin), that is unrelated to its spatial motion. The existence of spin angular momentum is inferred from experiments, such as the Stern-Gerlach experiment, in which particles are observed to possess angular momentum that cannot be accounted for by orbital angular momentum alone.

The total angular momentum, J, combines both the spin and orbital angular momentum of a particle (or a system), namely

Orbital Angular Momentum, Hydrogen Atom (Part - 1) - Angular Momentum, CSIR-NET Physical Sciences Physics Notes | EduRev


Orbital angular momentum 

Consider a particle of mass m, momentumOrbital Angular Momentum, Hydrogen Atom (Part - 1) - Angular Momentum, CSIR-NET Physical Sciences Physics Notes | EduRev and position vector Orbital Angular Momentum, Hydrogen Atom (Part - 1) - Angular Momentum, CSIR-NET Physical Sciences Physics Notes | EduRev (with respect to a fixed origin, Orbital Angular Momentum, Hydrogen Atom (Part - 1) - Angular Momentum, CSIR-NET Physical Sciences Physics Notes | EduRev = 0). In classical mechanics, the particle’s orbital angular momentum is given by a vector Orbital Angular Momentum, Hydrogen Atom (Part - 1) - Angular Momentum, CSIR-NET Physical Sciences Physics Notes | EduRev , defined by

Orbital Angular Momentum, Hydrogen Atom (Part - 1) - Angular Momentum, CSIR-NET Physical Sciences Physics Notes | EduRev. (1)

This vector points in a direction that is perpendicular to the plane containing Orbital Angular Momentum, Hydrogen Atom (Part - 1) - Angular Momentum, CSIR-NET Physical Sciences Physics Notes | EduRev, and has a magnitude L = rp sin α, where α is the angle between Orbital Angular Momentum, Hydrogen Atom (Part - 1) - Angular Momentum, CSIR-NET Physical Sciences Physics Notes | EduRev. In Cartesian coordinates, the components of Orbital Angular Momentum, Hydrogen Atom (Part - 1) - Angular Momentum, CSIR-NET Physical Sciences Physics Notes | EduRev are

Orbital Angular Momentum, Hydrogen Atom (Part - 1) - Angular Momentum, CSIR-NET Physical Sciences Physics Notes | EduRev(2)

Orbital Angular Momentum, Hydrogen Atom (Part - 1) - Angular Momentum, CSIR-NET Physical Sciences Physics Notes | EduRev

The corresponding QM operators representing Lx, Land Lz are obtained by replacing x, y, z and px, py and pz with the corresponding QM operators, giving

Orbital Angular Momentum, Hydrogen Atom (Part - 1) - Angular Momentum, CSIR-NET Physical Sciences Physics Notes | EduRev(3)

Orbital Angular Momentum, Hydrogen Atom (Part - 1) - Angular Momentum, CSIR-NET Physical Sciences Physics Notes | EduRev

In a more compact form, this can be written as a vector operator,

Orbital Angular Momentum, Hydrogen Atom (Part - 1) - Angular Momentum, CSIR-NET Physical Sciences Physics Notes | EduRev (4)

It is easy to verify that Orbital Angular Momentum, Hydrogen Atom (Part - 1) - Angular Momentum, CSIR-NET Physical Sciences Physics Notes | EduRev is Hermitian.
Using the commutation relations derived for Orbital Angular Momentum, Hydrogen Atom (Part - 1) - Angular Momentum, CSIR-NET Physical Sciences Physics Notes | EduRev, the commutation relations between the different components of Orbital Angular Momentum, Hydrogen Atom (Part - 1) - Angular Momentum, CSIR-NET Physical Sciences Physics Notes | EduRev are readily derived. For example:

Orbital Angular Momentum, Hydrogen Atom (Part - 1) - Angular Momentum, CSIR-NET Physical Sciences Physics Notes | EduRev

Since y and px commute with each other and with z and pz , the first term reads

Orbital Angular Momentum, Hydrogen Atom (Part - 1) - Angular Momentum, CSIR-NET Physical Sciences Physics Notes | EduRev (6)

Similarly, the second commutator gives

Orbital Angular Momentum, Hydrogen Atom (Part - 1) - Angular Momentum, CSIR-NET Physical Sciences Physics Notes | EduRev(7)

The third and forth commutators vanish; we thus find that

Orbital Angular Momentum, Hydrogen Atom (Part - 1) - Angular Momentum, CSIR-NET Physical Sciences Physics Notes | EduRev (8)

In a similar way, it is straightforward to show that

Orbital Angular Momentum, Hydrogen Atom (Part - 1) - Angular Momentum, CSIR-NET Physical Sciences Physics Notes | EduRev  (9)

and

Orbital Angular Momentum, Hydrogen Atom (Part - 1) - Angular Momentum, CSIR-NET Physical Sciences Physics Notes | EduRev (10)

The three equations are equivalent to the vectorial commutation relation:

Orbital Angular Momentum, Hydrogen Atom (Part - 1) - Angular Momentum, CSIR-NET Physical Sciences Physics Notes | EduRev   (11)

Note that this can only be true for operators; since, for regular vectors, clearly Orbital Angular Momentum, Hydrogen Atom (Part - 1) - Angular Momentum, CSIR-NET Physical Sciences Physics Notes | EduRev = 0.

The fact that the operators representing the different components of the angular momentum do not commute, implies that it is impossible to obtain definite values for all component of the angular momentum when measured simultaneously. This means that if the system is in eigenstate of one component of the angular momentum, it will in general not be an eigenstate of either of the other two components.
We define the operator representing the square of the magnitude of the orbital angular momentum by

Orbital Angular Momentum, Hydrogen Atom (Part - 1) - Angular Momentum, CSIR-NET Physical Sciences Physics Notes | EduRev (12)

It is easy to show that Orbital Angular Momentum, Hydrogen Atom (Part - 1) - Angular Momentum, CSIR-NET Physical Sciences Physics Notes | EduRev does commute with each of the three components: Lx, Ly or Lz .
For example (using Orbital Angular Momentum, Hydrogen Atom (Part - 1) - Angular Momentum, CSIR-NET Physical Sciences Physics Notes | EduRev= 0):

Orbital Angular Momentum, Hydrogen Atom (Part - 1) - Angular Momentum, CSIR-NET Physical Sciences Physics Notes | EduRev

Similarly,

Orbital Angular Momentum, Hydrogen Atom (Part - 1) - Angular Momentum, CSIR-NET Physical Sciences Physics Notes | EduRev(14)

which can be summarized as

Orbital Angular Momentum, Hydrogen Atom (Part - 1) - Angular Momentum, CSIR-NET Physical Sciences Physics Notes | EduRev (15)

Physically, this means that one can find simultaneous eigenfunctions of Orbital Angular Momentum, Hydrogen Atom (Part - 1) - Angular Momentum, CSIR-NET Physical Sciences Physics Notes | EduRev and one of the components of Orbital Angular Momentum, Hydrogen Atom (Part - 1) - Angular Momentum, CSIR-NET Physical Sciences Physics Notes | EduRev , implying that both the magnitude of the angular momentum and one of its components can be precisely determined. Once these are known, they fully specify the angular momentum.
In order to obtain the eigenvalues of Orbital Angular Momentum, Hydrogen Atom (Part - 1) - Angular Momentum, CSIR-NET Physical Sciences Physics Notes | EduRev and one of the components of Orbital Angular Momentum, Hydrogen Atom (Part - 1) - Angular Momentum, CSIR-NET Physical Sciences Physics Notes | EduRev (typically, Lz ), it is convenient to express the angular momentum operators in spherical polar coordinates: r, θ, φ, rather than the Cartesian coordinates x, y, z . The spherical coordinates are related to the Cartesian ones via

x = r sin θ cos φ;
y = r sin θ sin φ;
z = r cos θ.                                              (16)

After some algebra, one gets:

Orbital Angular Momentum, Hydrogen Atom (Part - 1) - Angular Momentum, CSIR-NET Physical Sciences Physics Notes | EduRev

We thus find that the operators Lx, Ly , Lz and Orbital Angular Momentum, Hydrogen Atom (Part - 1) - Angular Momentum, CSIR-NET Physical Sciences Physics Notes | EduRev depend on θ and φ only, that is they are independent on the radial coordinate Orbital Angular Momentum, Hydrogen Atom (Part - 1) - Angular Momentum, CSIR-NET Physical Sciences Physics Notes | EduRev. All these operators therefore commute with any function of r,

Orbital Angular Momentum, Hydrogen Atom (Part - 1) - Angular Momentum, CSIR-NET Physical Sciences Physics Notes | EduRev(18)

Also, obviously, if a wavefunction depends only on r (but not on θ, φ) it can be simultaneously an eigenfunction of Lx, Ly , Lz and L2. In all cases, the corresponding eigenvalue will be 0. (This is the only exception to the rule that that eigenvalues of one component (e.g., Lx) cannot be simultaneously eigenfunctions of the two other components of L).

 

Eigenvalues and eigenfunctions of L2 and L

Let us find now the common eigenfunctions to L2 and Lz , for a single particle. The choice of Lz (rather than, e.g., Lx) is motivated by the simpler expression (see Equation 17).

Eigenvalues of Lz

Since, in spherical coordinates Lz depends only on φ, we can denote its eigenvalue by Orbital Angular Momentum, Hydrogen Atom (Part - 1) - Angular Momentum, CSIR-NET Physical Sciences Physics Notes | EduRev and the corresponding eigenfunctions by Φm(φ). We thus have: 

Orbital Angular Momentum, Hydrogen Atom (Part - 1) - Angular Momentum, CSIR-NET Physical Sciences Physics Notes | EduRev(19)
namely

Orbital Angular Momentum, Hydrogen Atom (Part - 1) - Angular Momentum, CSIR-NET Physical Sciences Physics Notes | EduRev(20)
The solutions to this equation are

Orbital Angular Momentum, Hydrogen Atom (Part - 1) - Angular Momentum, CSIR-NET Physical Sciences Physics Notes | EduRev(21)

This is satisfied for any value of m; however, physically we require the wave function to be single valued (alternatively: continuous), namely Φm(2π) = Φm(0), from which we find

Orbital Angular Momentum, Hydrogen Atom (Part - 1) - Angular Momentum, CSIR-NET Physical Sciences Physics Notes | EduRev(22)

This equation is satisfied for m = 0, ±1, ±2, ±3, .... The eigenvalues of the operator Lz are thus m~, with m being integer (positive or negative) or zero. The number m is called the magnetic quantum number, due to the role it plays in the motion of charged particles in magnetic fields.
This means, that when measuring the z-component of an orbital angular momentum, one can only obtain Orbital Angular Momentum, Hydrogen Atom (Part - 1) - Angular Momentum, CSIR-NET Physical Sciences Physics Notes | EduRev Since the choice of the z direction was arbitrary, we see that the component of the orbital angular momentum about any axis is quantized.
The wavefunctions Φm(φ) are orthonormal, namely

Orbital Angular Momentum, Hydrogen Atom (Part - 1) - Angular Momentum, CSIR-NET Physical Sciences Physics Notes | EduRev(23)

Furthermore, they form a complete set, namely every function f (φ) can be written as

Orbital Angular Momentum, Hydrogen Atom (Part - 1) - Angular Momentum, CSIR-NET Physical Sciences Physics Notes | EduRev (24)
where the coefficients aare C-numbers.

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