نظام كمي ذو حالتين

An electrically neutral silver atoms beam through Stern–Gerlach experiment's inhomogeneous magnetic field splits into two, each of which corresponds to one possible spin value of the outermost electron of the silver atom.

In quantum mechanics, a two-state system (also known as a two-level system) is a quantum system that can exist in any quantum superposition of two independent (physically distinguishable) quantum states. The Hilbert space describing such a system is two-dimensional. Therefore, a complete basis spanning the space will consist of two independent states. Any two-state system can also be seen as a qubit.

Two-state systems are the simplest quantum systems that can exist, since the dynamics of a one-state system is trivial (i.e. there is no other state the system can exist in). The mathematical framework required for the analysis of two-state systems is that of linear differential equations and linear algebra of two-dimensional spaces. As a result, the dynamics of a two-state system can be solved analytically without any approximation. The generic behavior of the system is that the wavefunction's amplitude oscillates between the two states.

A very well known example of a two-state system is the spin of a spin-1/2 particle such as an electron, whose spin can have values +ħ/2 or −ħ/2, where ħ is the reduced Planck constant.

The two-state system cannot be used as a description of absorption or decay, because such processes require coupling to a continuum. Such processes would involve exponential decay of the amplitudes, but the solutions of the two-state system are oscillatory.

الحلول التحليلية لطاقات الحالة المستقرة والاعتماد على الزمن

تمثيل

Supposing the two available basis states of the system are $|1⟩$ and $|2⟩$, then in general the state can be written as a superposition of these two states with probability amplitudes $c_1,c_2$:

$|\psi ⟩=c_1|1⟩+c_2|2⟩$

Since, the basis states are orthonormal, $⟨i|j⟩={\delta }_{ij}$ where $i,j\in 1,2$ and ${\delta }_{ij}$ is the Kronecker delta, so $c_i=⟨i|\psi ⟩$. These two complex numbers may be considered coordinates in a two-dimensional complex Hilbert space. Thus the state vector corresponding to the state $|\psi ⟩$ is

$|\psi ⟩\equiv \left(\begin{array}{c}⟨1|\psi ⟩\\ ⟨2|\psi ⟩\end{array}\right)=\left(\begin{array}{c}{c}_1\\ c_2\end{array}\right)=c_1\left(\begin{array}{c}1\\ 0\end{array}\right)+c_2\left(\begin{array}{c}0\\ 1\end{array}\right)=\mathbf{c}$

and the basis states correspond to the basis vectors, $|1⟩\equiv \left(\begin{array}{c}⟨1|1⟩\\ ⟨2|1⟩\end{array}\right)=\left(\begin{array}{c}1\\ 0\end{array}\right)$ and $|2⟩\equiv \left(\begin{array}{c}⟨1|2⟩\\ ⟨2|2⟩\end{array}\right)=\left(\begin{array}{c}0\\ 1\end{array}\right)$.

If the state $|\psi ⟩$ is normalized, the norm of the statevector is unity, i.e. ${\left|c_1\right|}^2+{\left|c_2\right|}^2=1$.

All observable physical quantities, such as energy, are associated with hermitian operators. In the case of energy and the corresponding Hamiltonian, $H$ this means $H_{ij}=⟨i\left|H\right|j⟩=⟨j\left|H\right|i{⟩}^{*}=H_{ji}^{*}$, i.e. $H_{11}$ and $H_{22}$ are real, and $H_{12}=H_{21}^{*}$. Thus these four matrix elements $H_{ij}$ produce a 2 $\times$ 2 hermitian matrix.

$\mathbf{H}=\left(\begin{array}{cc}⟨1\left|H\right|1⟩& ⟨1\left|H\right|2⟩\\ ⟨2\left|H\right|1⟩& ⟨2\left|H\right|2⟩\end{array}\right)=\left(\begin{array}{cc}{H}_{11}& H_{12}\\ H_{12}^{*}& H_{22}\end{array}\right)$.

The Time-independent schrodinger equation states that $H|\psi ⟩=E|\psi ⟩$, and substituting for $|\psi ⟩$ in terms of the basis states from above, and premultiplying both sides by $⟨1|$ or $⟨2|$ produces a system of two linear equations that can be written in matrix form

$\left(\begin{array}{cc}{H}_{11}& H_{12}\\ H_{12}^{*}& H_{22}\end{array}\right)\left(\begin{array}{c}{c}_1\\ c_2\end{array}\right)=E\left(\begin{array}{c}{c}_1\\ c_2\end{array}\right)$

or $\mathbf{H}\mathbf{c}=E\mathbf{c}$ which is a 2 $\times$ 2 matrix Eigenvalues and eigenvectors problem. Because of the hermiticity of $\mathbf{H}$ the eigenvalues are real, or rather vice versa it is the requirement that the energies are real that implies the hermiticity of $\mathbf{H}$. The eigenvectors represent the stationary states, i.e. those for whom the absolute magnitude of the squares of the probability amplitudes do not change with time.

القيم الذاتية للهاملتونية

The most general form of a 2 $\times$ 2 Hermitian matrix such as the Hamiltonian of a two-state system is given by

$\mathbf{H}=\left(\begin{array}{cc}{ϵ}_1& \beta -i\gamma \\ \beta +i\gamma & {ϵ}_2\end{array}\right)$

where ${ϵ}_1,{ϵ}_2,\beta$ and $\gamma$ are real numbers with units of energy. The allowed energy levels of the system, namely the eigenvalues of the Hamiltonian matrix, can be found in the usual way.

Alternatively, this matrix can be decomposed as,

$\mathbf{H}=\alpha \cdot {\sigma }_0+\beta \cdot {\sigma }_1+\gamma \cdot {\sigma }_2+\delta \cdot {\sigma }_3=\left(\begin{array}{cc}\alpha +\delta & \beta -i\gamma \\ \beta +i\gamma & \alpha -\delta \end{array}\right)$

Here, $\alpha =\frac{({ϵ}_1+{ϵ}_2)}{2}$ and $\delta =\frac{({ϵ}_1-{ϵ}_2)}{2}$ are real numbers. The matrix ${\sigma }_0$ is the 2 $\times$ 2 identity matrix and the matrices ${\sigma }_k$ $(k=1,2,3)$ are the Pauli matrices. This decomposition simplifies the analysis of the system, especially in the time-independent case where the values of $\alpha ,\beta ,\gamma$ and $\delta$ are constants.

The Hamiltonian can be written even more compactly as:

$\mathbf{H}=\alpha \cdot {\sigma }_0+\mathbf{r}\cdot \mathit{\sigma }$

The vector $\mathbf{r}$ is given by $(\beta ,\gamma ,\delta )$ and $\sigma$ is given by $({\sigma }_1,{\sigma }_2,{\sigma }_3)$. This representation simplifies the analysis of the time evolution of the system and is easier to use with other specialized representations such as the Bloch sphere.

If the two-state system's time-independent Hamiltonian $H$ is defined as above, then its eigenvalues are given by $E_{\pm }=\alpha \pm \left|\mathbf{r}\right|$. Evidently $\alpha$ is the average energy of the two levels, and the norm of $\mathbf{r}$ is the splitting between them. The corresponding eigenvectors are denoted $|+⟩$ and $|-⟩$.

الاهتماد على الزمن

We now assume that the probability amplitudes are time-dependent, though the basis states are not. The Time-dependent Schrödinger equation states $i\hslash \frac{\partial }{\partial t}|\psi ⟩=H|\psi ⟩$, and proceeding as before (substituting for $|\psi ⟩$ and premultiplying by $⟨1|,⟨2|$ again produces a pair of coupled linear equations, but this time they are first order partial differential equations: $i\hslash \frac{\partial }{\partial t}\mathbf{c}=\mathbf{H}\mathbf{c}$. If $\mathbf{H}$ is time independent there are several approaches, to find the time dependence of $c_1,c_2$, such as normal modes. The result is that

$\mathbf{c}(t)=e^{-i\mathbf{H}t/\hslash }{\mathbf{c}}_0=\mathbf{U}(t){\mathbf{c}}_0$.

where ${\mathbf{c}}_0=\mathbf{c}(0)$ is the statevector at $t=0$. Here the exponential of a matrix may be found from the series expansion. The matrix $\mathbf{U}(t)$ is called the time evolution matrix (which comprises the matrix elements of the corresponding time evolution operator $U(t)$). It is easily proved that $\mathbf{U}(t)$ is unitary, meaning that ${\mathbf{U}}^{†}\mathbf{U}=1$. It can be shown that

$\mathbf{U}(t)=e^{-i\mathbf{H}t/\hslash }=e^{-i\alpha t/\hslash }(\cos (\left|\mathbf{r}\right|t/\hslash ){\sigma }_0-i\sin (\left|\mathbf{r}\right|t/\hslash )\hat{r}\cdot \mathit{\sigma })$

where $\hat{r}=\frac{\mathbf{r}}{\left|\mathbf{r}\right|}$.

When one changes the basis to the eigenvectors of the Hamiltonian, in other words, if the basis states $|1⟩,|2⟩$ are chosen to be the eigenvectors then ${ϵ}_1=H_{11}=⟨1\left|H\right|1⟩=E_1⟨1|1⟩=E_1$ and $\beta +i\gamma =H_{21}=⟨2\left|H\right|1⟩=E_1⟨2|1⟩=0$ and so the Hamiltonian is diagonal, i.e. $\left|\mathbf{r}\right|=\delta$ and is of the form,

$\mathbf{H}=\left(\begin{array}{cc}{E}_1& 0\\ 0& E_2\end{array}\right);$

Now the unitary time evolution operator $U$ is easily seen to be given by:

$\mathbf{U}(t)=e^{-i\mathbf{H}t/\hslash }=\left(\begin{array}{cc}{e}^{-i{E}_{1}t/\hslash }& 0\\ 0& e^{-i{E}_{2}t/\hslash }\end{array}\right)=e^{-i\alpha t/\hslash }\left(\begin{array}{cc}{e}^{-i\delta t/\hslash }& 0\\ 0& e^{i\delta t/\hslash }\end{array}\right)=e^{-i\alpha t/\hslash }(\cos (\delta t/\hslash ){\sigma }_0-i\sin (\delta t/\hslash ){\mathit{\sigma }}_3)$

The $e^{-i\alpha t/\hslash }$ factor only contributes to the overall phase of the operator and can usually be ignored to yield a new time evolution operator that is physically indistinguishable from the original operator. Moreover, any perturbation to the system (which will be of the same form as the Hamiltonian) can be added to the system in the eigenbasis of the unperturbed Hamiltonian and analysed in the same way as above. Therefore, for any perturbation the new eigenvectors of the perturbed system can be solved for exactly, as mentioned in the introduction.

صيغة رابي للاضطراب الاستاتيكي

Suppose that the system starts in one of the basis states at $t=0$, say $|1⟩$ so that ${\mathbf{c}}_0=\left(\begin{array}{c}1\\ 0\end{array}\right)$, and we are interested in the probability of occupation of each of the basis states as a function of time when $\mathbf{H}$ is the time-independent Hamiltonian.

$\mathbf{c}(t)=\mathbf{U}(t){\mathbf{c}}_0=\left(\begin{array}{cc}{U}_{11}(t)& U_{12}(t)\\ U_{21}(t)& U_{22}(t)\end{array}\right)\left(\begin{array}{c}1\\ 0\end{array}\right)=\left(\begin{array}{c}{U}_{11}(t)\\ U_{21}(t)\end{array}\right)$

The probability of occupation of state $i$ is $P_i(t)=|c_i(t){|}^2=|U_{i1}(t){|}^2$. In the case of the starting state, $P_1(t)=|c_1(t){|}^2=|U_{11}(t){|}^2$, and from above, $U_{11}(t)=e^{\frac{-i\alpha t}{\hslash }}(\cos (\left|\mathbf{r}\right|t/\hslash )-i\sin (\left|\mathbf{r}\right|t/\hslash )\frac{\delta }{\left|\mathbf{r}\right|})$. Hence

$P_1(t)={\cos }^2(\mathrm{\Omega }t)+{\sin }^2(\mathrm{\Omega }t)\frac{{\mathrm{\Delta }}^2}{{\mathrm{\Omega }}^2}$

Obviously $P_1(0)=1$ due to the initial condition. The frequency $\mathrm{\Omega }=\left|\mathbf{r}\right|/\hslash =\sqrt{{\beta }^2+{\gamma }^2+{\delta }^2}/\hslash =\sqrt{|{\mathrm{\Omega }}_R{|}^2+{\mathrm{\Delta }}^2}$ is called the generalised Rabi frequency, ${\mathrm{\Omega }}_R=(\beta +i\gamma )/\hslash$ is called the Rabi frequency, and $\mathrm{\Delta }=\delta /\hslash$ is called the detuning. At zero detuning, $P_1(t)={\cos }^2(|{\mathrm{\Omega }}_R|t)$, i.e. there is Rabi flopping from guaranteed occupation of state 1, to guaranteed occupation of state 2, and back to state 1 etc with frequency $\left|{\mathrm{\Omega }}_R\right|$. As the detuning is increased away from zero, the frequency of the flopping increases (to $\mathrm{\Omega }$) and the amplitude decreases to $1-{\mathrm{\Delta }}^2/{\mathrm{\Omega }}^2$.

See also Rabi cycle and Rotating wave approximation for time dependent Hamiltonians induced by light waves.

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