Matrix Operators and The Two-Level System

We have already seen how confinement of particles leads to discrete energies $E_n$ and wave functions localized to certain regions of space. Furthermore, we have learned that the coupling of two separated quantum systems (like the potential wells in section 2.6) leads to the splitting of energy levels. The tunnel effect couples the left and the right well and leads to new states that are superpositions, cf. (2.89),

$$\psi_{\pm}(x) = \psi_{L}(x) \pm \psi_{R}(x).$$ (184)

We had seen furthermore that the derivation of this result, starting from the one-dimensional Schrödinger equation, is quite lengthy, because transcendental equations for the possible energies have to be solved at least approximatively. We therefore would like to find a simpler, slightly more abstract model, that describes the main physics of the level splitting and the tunnel effect, leading to eigenstates like the $\psi_{\pm}$ above.

We consider again a system were a particle is moving in a potential that has the form of a double well like the one in section 2.6. We are interested in the case where the barrier between the two wells is very high. Let us concentrate on the wave functions with the lowest energies. We know already that we can express them approximately by the linear combinations $\psi_{\pm}$ of the two lowest states $\psi_{L}(x)$ and $\psi_{R}(x)$ of the left and the right well.

Figure: Double well potential (left) and its two lowest states.

We now perform the 5 steps that establish a simple model of what is going on when the two wells become coupled by the barrier:

STEP 1: starting from two isolated wells, we completely neglect all states apart from the two ground states in the two wells, $\psi_{L}(x)$ and $\psi_{R}(x)$. We are only interested in the `low-energy' sector. If both wells have a small width, we know that the next eigenvalue of the energy is far above the ground state energy so that all other states are energetically far away from the two ground states $\psi_{L}(x)$ and $\psi_{R}(x)$.

STEP 2: we now define these two ground states as the two basis vectors of a complex two-dimensional Hilbert space ${\cal H} = C^2$. We try to discuss all the following quantum mechanical features within this `small' Hilbert space which shall be our approximation of what `really is going on'.

STEP 3: we call the basis vectors $\vert L\rangle$ (corresponding to the wave function $\psi_{L}(x)$) and $\vert R\rangle$ (corresponding to the wave function $\psi_{R}(x)$). We consider $\vert R\rangle$ and $\vert L\rangle$ just as basis vectors of $C^2$. The particular form of the corresponding wave functions does not interest us. We rather introduce the notation for basis vectors known from linear algebra, that is

$$\vert L\rangle = \left(\begin{array}{c} 1 \\ 0 \end{array}\right),\quad \vert R\rangle = \left(\begin{array}{c} 0 \\ 1 \end{array}\right).$$ (185)

The scalar product in ${\cal H} = C^2$ is the standard scalar product for vectors: although the basis vectors $\vert L\rangle$ and $\vert R\rangle$ correspond to the two wave functions in the left and right well, here they are really vectors. In our abstract model we don't care a hell about what $\vert L\rangle$ and $\vert R\rangle$ stand for.

Figure: Vector representation of left and right lowest states of double well potential.

STEP 4: We associate a Hamiltonian $\hat{H_0}$ with the two isolated wells: trivially, the particle is either in the left or in the right well. A measurement of the energy (the observable belonging to $\hat{H_0}$) yields one of the eigenvalues of $\hat{H_0}$, i.e $E_L$ (the energy of the lowest state left) or $E_R$ (the energy of the lowest state right). In fact, in section 2.6 we always had $E_L=E_R$ but let us be a bit more general here and allow different ground state energies in both isolated wells. The Hamiltonian is a two-by-two matrix,

$$\hat{H}_0=\left( \begin{array}{cc} E_L & 0\\ 0 & E_R \end{array}\right)$$ (186)

because with this form

$$\hat{H}_0 \vert L\rangle = E_L \vert L\rangle, \quad \hat{H}_0 \vert R\rangle = E_R \vert R\rangle,$$ (187)

that is $\vert L\rangle$ is eigenvector with eigenvalue $E_L$ and $\vert R\rangle$ is eigenvector with eigenvalue $E_R$.

STEP 5 (this is the most abstract step): we now want to incorporate the tunnel effect when the two wells become coupled by a barrier of finite height. What is the total Hamiltonian $\hat{H}$ of the system then? A particle initially localized in the left well can now tunnel into the right well and vice versa. The time-evolution of the wave function is determined by the total Hamiltonian (remember the time-dependent Schrödinger equation!) which therefore must contain a term like

$$\hat{T}:=\left( \begin{array}{cc} 0 & T\\ T^* & 0 \end{array... ...vert R \rangle,\quad \hat{T}\vert R\rangle = T \vert L \rangle.$$ (188)

The operator $\hat{T}$ changes $\vert L\rangle$ into $\vert R\rangle$ and $\vert R\rangle$ into $\vert L\rangle$, i.e. it puts the particle from the left to the right and from the right to the left which mimics the tunnel process. The strength of this process is proportional to $T$ which is a free complex parameter in this model.

Furthermore, the energies $E_L$ and $E_R$ are changed: we therefore write the total Hamiltonian as a sum of three terms,

$$\hat{H}=\left( \begin{array}{cc} E_L & 0\\ 0 & E_R \end{arra... ...c} \varepsilon_L & T\\ T^* & \varepsilon_R \end{array}\right).$$ (189)

We check first that $\hat{H}$ is hermitian as it must be: this is the reason why we have $T^*$ as $\hat{H}_{21}$.

The two-by-two matrix Hamiltonian $\hat{H}$, (3.24), is called the Hamiltonian of the two-level system. It describes the simplest possible quantum mechanical system in terms of the three parameters $\varepsilon_L$, $\varepsilon_R$, and $T$. In spite of its simplicity, this model is the basis for a lot of phenomena in different fields of physics, such as the dynamics of emission and absorption of light from atoms, the nuclear magnetic resonance (NMR), the spin $1/2$ of particles, the physics of semiconductors with two bands (valence and conduction band), and many others.