Angular momentum (quantum): Difference between revisions
imported>Paul Wormer |
imported>Paul Wormer |
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It can be shown from the above definitions that '''j'''<sup>2</sup> commutes with ''j''<sub>''x''</sub>, ''j''<sub>''y''</sub>, and ''j''<sub>''z''</sub> | It can be shown from the above definitions that '''j'''<sup>2</sup> commutes with ''j''<sub>''x''</sub>, ''j''<sub>''y''</sub>, and ''j''<sub>''z''</sub> | ||
:<math> | :<math> | ||
[\mathbf{j}^2, j_k] = 0 \quad \mathrm{for}\;\; k = x,y,z. | [\mathbf{j}^2,\, j_k] = 0 \quad \mathrm{for}\;\; k = x,y,z. | ||
</math> | </math> | ||
When two Hermitian operators commute a common set of eigenfunctions exists. | When two Hermitian operators commute a common set of eigenfunctions exists. | ||
Conventionally | Conventionally ''j''<sub>''z''</sub> is chosen to supplement '''j'''<sup>2</sup>. | ||
From the commutation relations the possible eigenvalues can be found. | From the commutation relations the possible eigenvalues can be found. | ||
The result is | The result is | ||
| Line 150: | Line 150: | ||
C_\pm(j,m) = \sqrt{j(j+1)-m(m\pm 1)} = \sqrt{(j\mp m)(j\pm m + 1)}. | C_\pm(j,m) = \sqrt{j(j+1)-m(m\pm 1)} = \sqrt{(j\mp m)(j\pm m + 1)}. | ||
</math> | </math> | ||
A (complex) phase factor could be included in the definition of <math>C_\pm(j,m)</math> | A (complex) phase factor could be included in the definition of <math>\scriptstyle C_\pm(j,m)</math> | ||
The choice made here is in agreement with the Condon and Shortley phase conventions. | The choice made here is in agreement with the Condon and Shortley phase conventions. | ||
The angular momentum states must be orthogonal (because their eigenvalues with | The angular momentum states must be orthogonal (because their eigenvalues with | ||
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\langle j m | j' m' \rangle = \delta_{j,j'}\delta_{m,m'}. | \langle j m | j' m' \rangle = \delta_{j,j'}\delta_{m,m'}. | ||
</math> | </math> | ||
===Proof of properties of eigenstates=== | |||
The angular momentum operators satisfy | |||
:<math> | |||
[\mathbf{j}^2,\, j_\pm] = [\mathbf{j}^2,\, j_z]= 0,\qquad [j_z,\,j_\pm] = \pm j_\pm,\qquad j_-j_+ = \mathbf{j}^2 - j_z(j_z+1). | |||
</math> | |||
The main steps in the construction of the eigenstates are: | |||
* Since '''j'''<sup>2</sup> and ''j''<sub>''z''</sub> commute, we can find a common eigenvector <math> \scriptstyle |a,b\rangle\, </math> with | |||
:<math> | |||
\mathbf{j}^2 |a,b\rangle\, =\, a^2|a,b\rangle\quad \hbox{and}\quad j_z |a,b\rangle\, =\, b|a,b\rangle | |||
</math>. | |||
Since a Hermitian operator squared has only real, nonnegative, [[expectation values]], <math>\scriptstyle \langle \psi| A^2 | \psi \rangle\, =\, \langle A\psi| A \psi \rangle \ge 0 </math>, and since an eigenvalue is a special kind of expectation value—namely one with respect to an eigenvector—it follows that '''j'''<sup>2</sup> has only non-negative real eigenvalues. Therefore we write its eigenvalue as a the squared number <math>\scriptstyle a^2</math>. | |||
<!-- | |||
\item{} | |||
Considering the commutation relations $[j_{\pm}, j_3] = \pm j_{\pm}$ | |||
and $[j^2, j_{\pm}]=0$, we find, | |||
that $j^2 j_+\ket{a,b} = a^2 j_+\ket{a,b} $ and | |||
$j_3 j_+ \ket{a,b} = (b+1)j_+\ket{a,b}$. Hence | |||
$j_+\ket{a,b} = \ket{a,b+1}$ | |||
\item{} If we apply $j_+$ now $k+1$ times we obtain, using | |||
$j_+^\dagger = j_-$, the ket | |||
$\ket{a,b+k+1}$ with norm | |||
\begin{equation} | |||
\braopket{a,b+k}{j_-j_+}{a,b+k}= | |||
[a^2-(b+k)(b+k+1)] \braket{a,b+k}{a,b+k}. | |||
\end{equation} | |||
Thus, if we let $k$ increase, there comes a point that the norm | |||
on the left hand side would have to be negative (or zero), | |||
while the norm on the right hand side would still be positive. | |||
A negative norm is in contradiction with the fact that the ket | |||
belongs to the Hilbert space ${\cal L}$. Hence there must exist a value | |||
of the integer $k$, such that the ket $\ket{a,b+k} \ne 0$, | |||
while $\ket{a,b+k+1} = 0 $. Also $a^2=(b+k)(b+k+1)$ for that value of | |||
$k$. | |||
\item{} Similarly $l+1$ times application of $j_-$ | |||
gives a zero ket $\ket{a,b-l-1}$ with $\ket{a,b-l} \ne 0$ and | |||
$a^2=(b-l)(b-l-1)$. | |||
\item{} From the fact that $a^2=(b+k)(b+k+1)=(b-l)(b-l-1)$ | |||
follows $2b=l-k$, so that $b$ is integer or half-integer. This quantum | |||
number is traditionally designated by $m$. The maximum value of $m$ | |||
will be designated by $j$. Hence $a^2=j(j+1)$. | |||
\item{} Requiring that $\ket{j,m}$ and $j_\pm \ket{j,m}$ are normalized | |||
and fixing phases, we obtain the well-known formula (\ref{updown}). | |||
\end{itemize} | |||
The spaces with half-integer $j$ (and $m$) belong to a spin Hilbert | |||
space. In agreement with what was stated in the introduction, we will | |||
not consider half-integer $j$ any further. | |||
--> | |||
==References== | ==References== | ||
<references /> | <references /> | ||
'''(to be continued)''' | '''(to be continued)''' | ||
Revision as of 08:04, 27 December 2007
In quantum mechanics, angular momentum is a vector operator of which the three components have well-defined commutation relations. This operator is the quantum analogue of the classical angular momentum vector.
Angular momentum entered quantum mechanics in one of the very first—and most important—papers on the "new" quantum mechanics, the Dreimännerarbeit (three men's work) of Born, Heisenberg and Jordan (1926).[1] In this paper the orbital angular momentum and its eigenstates are already fully covered by the algebraic techniques of commutation relations and step up/down operators that will be treated in the present article. In 1927, Wolfgang Pauli introduced spin angular momentum,[2] which is a form of angular momentum without a classical counterpart.
Angular momentum theory—together with its connection to group theory— brought order to a bewildering number of spectroscopic observations in atomic spectroscopy, see, for instance, Wigner's seminal work.[3] When in 1926 electron spin was discovered and Pauli proved less than a year later that spin was a form of angular momentum, its importance rose even further. To date the theory of angular momentum is of great importance in quantum mechanics. It is an indispensable discipline for the working physicist, irrespective of his field of specialization, be it solid state physics, molecular-, atomic,- nuclear,- or even hadronic-structure physics.[4]
Orbital angular momentum
The classical angular momentum of a point mass is,
where r is the position and p the (linear) momentum of the point mass. The simplest and oldest example of an angular momentum operator is obtained by applying the quantization rule:
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \hbar } is Planck's constant (divided by 2π) and ∇ is the gradient operator. This rule applied to the classical angular momentum vector gives a vector operator with the following three components,
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} L_x &= -i\hbar\Big( y \frac{\partial}{\partial z} - z \frac{\partial}{\partial y}\Big) \\ L_y &= -i\hbar\Big( z \frac{\partial}{\partial x} - x \frac{\partial}{\partial z}\Big) \\ L_z &= -i\hbar\Big( x \frac{\partial}{\partial y} - y \frac{\partial}{\partial x}\Big). \\ \end{align} }
The following commutation relations can be proved,
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [L_x,\,L_y] = i \hbar L_z, \quad [L_z,\,L_x] = i \hbar L_y, \quad [L_y,\,L_z] = i \hbar L_x. }
The square brackets indicate the commutator of two operators, defined for two arbitrary operators A and B as
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [A,\,B] \equiv AB - BA . }
For instance,
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \big[L_x,\, L_y\big] =& -\hbar^2\left[ \Big( y \frac{\partial}{\partial z} - z \frac{\partial}{\partial y}\Big) \Big( z \frac{\partial}{\partial x} - x \frac{\partial}{\partial z}\Big) - \Big( z \frac{\partial}{\partial x} - x \frac{\partial}{\partial z}\Big)\Big( y \frac{\partial}{\partial z} - z \frac{\partial}{\partial y}\Big) \right] \\ =& -\hbar^2\left[ y \frac{\partial}{\partial x} - x \frac{\partial}{\partial y} \right] = i \hbar \left[-i\hbar \Big( x \frac{\partial}{\partial y} - y \frac{\partial}{\partial x} \Big)\right] = i\hbar L_z, \\ \end{align} }
where we used that all the terms of the kind
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle yx \frac{\partial^2}{\partial z \partial x}, \quad \hbox{etc.}}
mutually cancel.
The total angular momentum squared is defined by
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{L}^2 \equiv L_x^2 +L_y^2 +L_z^2. }
In terms of spherical polar coordinates the operator is,
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L^2 = - \left[ \frac{1}{\sin\theta} \frac{\partial}{\partial\theta} \sin\theta \frac{\partial}{\partial \theta} + \frac{1}{\sin^2\theta} \frac{\partial^2}{\partial\varphi^2}\right]. }
Note, parenthetically, that eigenfunctions of the latter operator have been known since the nineteenth century, long before quantum mechanics was born. They are spherical harmonic functions.
Spin angular momentum
Pauli introduced in 1927 the following three matrices, which are now known as Pauli spin matrices,
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\sigma}_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \\ \end{pmatrix}, \qquad \boldsymbol{\sigma}_y = \begin{pmatrix} 0 & -i \\ i & 0 \\ \end{pmatrix}, \qquad \boldsymbol{\sigma}_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \\ \end{pmatrix}. }
These Hermitian matrices represent Hermitian operators on a two-dimensional linear space over the field of complex numbers: spin space. Spin angular momentum operators are defined by
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s_x \leftrightarrow \frac{\hbar}{2}\boldsymbol{\sigma}_x\qquad s_y \leftrightarrow \frac{\hbar}{2}\boldsymbol{\sigma}_y\qquad s_z \leftrightarrow \frac{\hbar}{2}\boldsymbol{\sigma}_z. }
The commutation relations of these operators follow by matrix multiplication, for instance,
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [s_x,\,s_y] \leftrightarrow \frac{\hbar^2}{4}[\boldsymbol{\sigma}_x\boldsymbol{\sigma}_y -\boldsymbol{\sigma}_y\boldsymbol{\sigma}_x] = \frac{i\hbar^2}{2} \begin{pmatrix} 1 & 0 \\ 0 & -1 \\ \end{pmatrix} \leftrightarrow i \hbar s_z. }
It is shown in this manner that
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [s_x,\,s_y] = i \hbar s_z, \quad [s_z,\,s_x] = i \hbar s_y, \quad [s_y,\,s_z] = i \hbar s_x, }
which may be compared with the commutation relations of the orbital angular momenta given earlier.
Abstract angular momentum operators
We have seen two examples of angular momentum operators, but many more can be given. For instance, the sum operator s + L, or sum operators of more than one particle are also angular momentum operators. The essential characteristic that all these operator share is that they have three components with well-defined commutation relations. Taking a somewhat more abstract point of view, one comes to the following definition: An angular momentum operator is a vector operator with three Hermitian component operators jx, jy, and jz, that satisfy the commutation relations
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [j_k,j_l] = i \hbar\sum_{m=x,y,z} \varepsilon_{klm}j_m, \quad k, l, m = x,y,z, }
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon_{klm}} is the Levi-Civita symbol,
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon_{klm} = \begin{cases} 0 & \hbox{if two or more indices are zero} \\ 1 & \hbox{if }k l m \hbox{ is an even permutation of 1 2 3} \\ -1 & \hbox{if }k l m \hbox{ is an odd permutation of 1 2 3} \\ \end{cases} }
Together the three components define the vector operator j. The square of the length of j is defined as
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{j}^2 = j_x^2+j_y^2+j_z^2. }
We also define raising j+ and lowering j− operators (also known as step up/down operators),
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j_\pm = j_x \pm i j_y. \, }
Angular momentum states
It can be shown from the above definitions that j2 commutes with jx, jy, and jz
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [\mathbf{j}^2,\, j_k] = 0 \quad \mathrm{for}\;\; k = x,y,z. }
When two Hermitian operators commute a common set of eigenfunctions exists. Conventionally jz is chosen to supplement j2. From the commutation relations the possible eigenvalues can be found. The result is
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{j}^2 |j m\rangle = j(j+1) |j m\rangle, \qquad j=0, 1/2, 1, 3/2, 2, \ldots }
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j_z|j m\rangle = m |j m\rangle, \qquad\quad m = -j, -j+1, \ldots , j. }
The raising and lowering operators change the value of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j_\pm |jm\rangle = C_\pm(j,m) |j m\pm 1\rangle }
with
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_\pm(j,m) = \sqrt{j(j+1)-m(m\pm 1)} = \sqrt{(j\mp m)(j\pm m + 1)}. }
A (complex) phase factor could be included in the definition of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle C_\pm(j,m)} The choice made here is in agreement with the Condon and Shortley phase conventions. The angular momentum states must be orthogonal (because their eigenvalues with respect to a Hermitian operator are distinct) and they are assumed to be normalized
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle j m | j' m' \rangle = \delta_{j,j'}\delta_{m,m'}. }
Proof of properties of eigenstates
The angular momentum operators satisfy
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [\mathbf{j}^2,\, j_\pm] = [\mathbf{j}^2,\, j_z]= 0,\qquad [j_z,\,j_\pm] = \pm j_\pm,\qquad j_-j_+ = \mathbf{j}^2 - j_z(j_z+1). }
The main steps in the construction of the eigenstates are:
- Since j2 and jz commute, we can find a common eigenvector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle |a,b\rangle\, } with
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{j}^2 |a,b\rangle\, =\, a^2|a,b\rangle\quad \hbox{and}\quad j_z |a,b\rangle\, =\, b|a,b\rangle } .
Since a Hermitian operator squared has only real, nonnegative, expectation values, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \langle \psi| A^2 | \psi \rangle\, =\, \langle A\psi| A \psi \rangle \ge 0 } , and since an eigenvalue is a special kind of expectation value—namely one with respect to an eigenvector—it follows that j2 has only non-negative real eigenvalues. Therefore we write its eigenvalue as a the squared number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle a^2} .
References
- ↑ M. Born, W. Heisenberg, and P. Jordan, Zur Quantenmachanik II, Zeitschrift f. Physik. vol. 35, pp. 557-615 (1926)
- ↑ W. Pauli jr., Zur Quantenmechanik des magnetischen Elektrons, Zeitschrift f. Physik. vol. 43, pp. 601-623 (1927)
- ↑ E. P. Wigner, Gruppentheorie und ihre Anwendungen auf die Quantenmechanik der Atomspektren, Vieweg Verlag, Braunschweig (1931). Translated into English: J. J. Griffin, Group Theory and its Application to the Quantum Mechanics of Atomic Spectra Academic Press, New York (1959).
- ↑ L. C. Biedenharn, J. D. Louck, Angular Momentum in Quantum Physics, Addison-Wesley, Reading, Massachusetts (1981)
(to be continued)