Abstract
Up to now, we have characterized a quantum-mechanical system by a vector \(\left| \psi \right\rangle \) of the Hilbert space. In the following, we will extend the concept of state, as we already promised in Chap. 14, Vol. 1. We will introduce the density operator or density matrix, the most general representation of states in quantum mechanics. This tool allows us to describe also states for which we do not have complete information, and which therefore cannot be represented by a vector in the Hilbert space. That such a description is useful or necessary may be surprising at first, but we will see that this formulation is quite handy, especially with regard to the discussion of the measurement process in quantum mechanics.
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- 1.
A dependence on time is not mentioned explicitly in the following.
- 2.
Actually, it would be better to say ‘state operator’, but the term ‘density operator’ has become established—and unfortunately also the use of \(\rho \). Note: Previously, \(\rho \) was used exclusively for the probability density, i.e. \(\rho =\left\langle \psi \right| \left. \psi \right\rangle \). Which \(\rho \) is meant in a particular situation should follow unambiguously from the context.
- 3.
Indeed, states are strictly speaking not determined by vectors of \({\mathcal {H }}\), but instead by rays, see Chap. 14, Vol. 1.
- 4.
Gottfried and Yan, Quantum Mechanics: Fundamentals, p. 46.
- 5.
Another example for the quasi natural occurrence of the density operator is the theorem of Gleason, which deals with the question of how mean values (or probabilities) can be defined in quantum mechanics. It states in essence that in Hilbert spaces of dimension \({\ge }3\), the mean values of projection operators P can be described only by \(\left\langle P\right\rangle =tr\left( \rho P\right) \) (see Appendix T, Vol. 2).
- 6.
See also Appendix F, Vol. 1.
- 7.
It holds that \(tr\left( \rho A\right) =tr\left( A\rho \right) \); see the exercises.
- 8.
The density matrix is Hermitian, of course.
- 9.
Since the density matrix is Hermitian, its eigenvalues are real.
- 10.
This does not change if we diagonalize the matrix (that is possible, since it is Hermitian). For the diagonal elements of the resulting diagonal matrix are the eigenvalues, and in this case they are 0 and 1 (because of \( \left| c_{1}\right| ^{2}+\left| c_{2}\right| ^{2}=1\)); see the exercises.
- 11.
In contrast to the probabilities which are inevitable in quantum mechanics.
- 12.
For the expansion in terms of not-necessarily-orthogonal states, see the exercises.
- 13.
The case that all the probabilities are zero is trivial.
- 14.
Note: For a change of basis to another CONS \(\left\{ \left| \varphi _{m}\right\rangle \right\} \) with
$$\begin{aligned} \left| n\right\rangle =\sum _{m}c_{nm}\left| \varphi _{m}\right\rangle , \end{aligned}$$off-diagonal terms can of course occur in the density matrix:
$$\begin{aligned} \rho _\mathrm{stat}=\sum \limits _{n,m,m^{\prime }}p_{n}c_{nm}c_{n^{\prime }m^{\prime }}^{*}\left| \varphi _{m}\right\rangle \left\langle \varphi _{m^{\prime }}\right| =\sum \limits _{m,m^{\prime }}\sum _{n}\left( p_{n}c_{nm}c_{n^{\prime }m^{\prime }}^{*}\right) \left| \varphi _{m}\right\rangle \left\langle \varphi _{m^{\prime }}\right| =\sum \limits _{m, m^{\prime }}\alpha _{mm^{\prime }}\left| \varphi _{m}\right\rangle \left\langle \varphi _{m^{\prime }}\right| . \end{aligned}$$The occurrence of non-trivial off-diagonal terms is not crucial as a distinguishing feature between \(\rho \) and \(\rho _\mathrm{stat}\), but rather the criteria \(\rho _\mathrm{stat}^{2}\ne \rho _\mathrm{stat}\) and \(tr\left( \rho _\mathrm{stat}^{2}\right) <1\) (thus the denotation of off-diagonal terms as ‘coherences’ or ‘interference terms’ is a bit unfortunate).
- 15.
What is precisely meant must follow from the context, if necessary.
- 16.
Compare this with the Liouville equation in classical mechanics:
$$\begin{aligned} \partial _{t}\rho =\left\{ H,\rho \right\} _\mathrm{Poisson} \end{aligned}$$where the Poisson bracket of two quantities A and B is defined by
$$\begin{aligned} \left\{ F, G\right\} _\mathrm{Poisson}=\mathop {\displaystyle \sum }\limits _{k}\left( \frac{\partial F}{\partial q_{k}}\frac{\partial G}{\partial p_{k}}-\frac{ \partial G}{\partial q_{k}}\frac{\partial F}{\partial p_{k}}\right) . \end{aligned}$$This motivates once more the transition from classical mechanics to quantum mechanics by the substitution (keyword canonical quantization; see Appendix W, Vol. 2)
$$\begin{aligned} \left\{ ,\right\} _{\text {Poisson bracket}}\rightarrow \frac{1}{i\hbar } \left[ ,\right] _{\text {commutator}} \end{aligned}$$.
- 17.
We note that \(\left\langle m_{2}\right| \rho \left| m_{2}\right\rangle \) is an operator in \({\mathcal {H}}_{1}\), and therefore, it generally holds that \(\ \sum \limits _{m_{2}}\left\langle m_{2}\right| \rho \left| m_{2}\right\rangle A_{1}\ne A_{1}\sum \limits _{m_{2}}\left\langle m_{2}\right| \rho \left| m_{2}\right\rangle \).
- 18.
Other notations are \( \hat{\rho }\), \(\rho _{red}\), \(\rho _{1}\) or similar forms.
- 19.
In this representation, we can see directly that \(\rho ^{(1)}\) is Hermitian and positive.
- 20.
In the following chapters, we will consider three or more quantum objects. The reduced density operator for quantum object 1 is then given by taking the trace over all other quantum objects (colloquially, ’tracing out’ these degrees of freedom).
- 21.
One can imagine e.g. two photons, and by \(\left| 1\right\rangle \) and \( \left| 2\right\rangle \) e.g. \(\left| h\right\rangle \) and \( \left| v\right\rangle \) or \(\left| r\right\rangle \) and \(\left| l\right\rangle \).
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Pade, J. (2018). The Density Operator. In: Quantum Mechanics for Pedestrians 2. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-00467-5_22
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DOI: https://doi.org/10.1007/978-3-030-00467-5_22
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