15. Wave quantum mechanics

If the system behaves like a wave, it must satisfy a wave function

As the geometrical optics is the short wave-length approximation of the ondulatory optics, classical mechanics may be the short wave-length approximation of ondulatory mechanics

15.1. De Broglie

15.2. Schrodinger

15.2.1. Derivation

From the probability interpretation of the wave function \(| \Psi_t \rangle\), and the unitary condition \(\langle \Psi_t | \Psi_t \rangle = 1\), for every \(t\), the evolution of the system must be governed by a unitary operator \(U_{t,t_0}\),

\[ | \Psi_t \rangle = U_{t,t_0} | \Psi_{t_0} \rangle \ ,\]

for the unitary condition to hold at every \(t\),

\[1 = \langle \Psi_{t} | \Psi_{t} \rangle = \langle U_{t,t_0} \Psi_{t_0} | U_{t,t_0} \Psi_{t_0} \rangle = \langle \Psi_{t_0} | \underbrace{U^{\dagger}_{t,t_0} U_{t,t_0}}_{= \hat{\mathbf{1}}} | \Psi_{t_0} \rangle = \langle \Psi_{t_0} | \Psi_{t_0} \rangle \ ,\]

i.e.

\[U_{t,t_0}^{-1} = U^{\dagger}_{t,t_0} \ .\]

Taking the time derivative w.r.t. \(t\) of the evolution relation, it follows

\[| \dot{\Psi}_t \rangle = \partial_t U_{t,t_0} | \Psi_{t_0} \rangle = \underbrace{ \partial_t U_{t,t_0} U^{\dagger}_{t,t_0}}_{ \hat{A}(t,t_0)} | \Psi_{t} \rangle \ .\]

Composition. As \(| \Psi_{t} \rangle = U_{t,t_1} | \Psi_{t_1} \rangle = U_{t,t_1} U_{t_1,t_0}| \Psi_{t_0} \rangle\),

\[U_{t,t_0} = U_{t,t_1} U_{t_1,t_0} \ .\]

Indpendence of \(\hat{A}(t,t_0) = \partial_t U_{t,t_0} U_{t,t_0}\) from \(t_0\). The operator \(\hat{A}(t,t_0) = \partial_t U_{t,t_0} U_{t,t_0}\) can thus be written as

\[\hat{A}(t,t_0) = \partial_t U_{t,t_0} U_{t,t_0}^{\dagger} = \partial_t U_{t,t_1} \underbrace{U_{t_1,t_0} U_{t_1, t_0}^{\dagger}}_{ = \hat{\mathbf{1}} } U_{t,t_1}^{\dagger} = \partial_t U_{t,t_1} U_{t,t_1}^{\dagger} = \hat{A}(t,t_1) \ ,\]

showing that it’s independent from the initial time, \(\hat{A}(t,t_0) = \hat{A}(t)\).

\(\hat{A}(t)\) is anti-Hermitian. As

\[0 = \partial_t \underbrace{\left( U_{t,t_0} U_{t,t_0}^{\dagger} \right)}_{= \hat{\mathbf{1}} } = \underbrace{ \partial_t U_{t,t_0} U_{t,t_0}^\dagger}_{ = \hat{A}(t)} + \underbrace{ U_{t,t_0} \, \partial_t U_{t,t_0}^\dagger}_{= \hat{A}^\dagger(t)} = \hat{A}(t) + \hat{A}^\dagger(t) \ ,\]

and thus \(\hat{A} = - \hat{A}^\dagger\). Thus the operator \(\hat{A}\) can be written as \(\hat{A} = i \hat{\tilde{H}}\), being \(\hat{\tilde{H}}\) and Hermitian operator. For the correspondence principle (classical limit for \(\hbar \rightarrow 0\)) the Hermitian operator is found to be \(\hat{\tilde{H}} = -\frac{1}{\hbar} \hat{H}\), being \(\hat{H}\) the Hamiltonian operator. todo Add a link to “correspondence principle”, either Ehrenfest theorem and/ or equations in Heisenberg picture

Thus, the wave equation becomes

(15.1)\[| \dot{\Psi} \rangle = \frac{1}{i \hbar} \hat{H} | \Psi \rangle \qquad \text{or} \qquad i \hbar | \dot{\Psi} \rangle = \hat{H} | \Psi \rangle \ .\]

The Hamiltonian can be written as a function of the unitary operator \(U_{t,t_0}\) and its time derivative as

(15.2)\[\hat{H} = i \hbar \, \partial_t U_{t,t_0} U^{\dagger}_{t,t_0} \ .\]

15.2.2. Energy eigen-states

\[\hat{H} | \Psi_k \rangle = E_k | \Psi_k \rangle \ .\]

Let the Hamiltonian be independent from time, thus also the eigenvalues and eigenfunctions are independent from time. Let a state be a superposition (linear combination) of eigenfunctions of the Hamiltonian operator

\[| \Psi_t \rangle = c_{k}(t) | \Psi_k \rangle \ ,\]

the time evolution immedately follows as

\[\begin{aligned} i \hbar \dot{c}_k | \Psi_k \rangle = c_k \hat{H} | \Psi_k \rangle = c_k E_k | \Psi_k \rangle \ , \end{aligned}\]

from orthogonality condition \(\langle \Psi_j | \Psi_k \rangle = \delta_{jk}\), that gives the dynamical equation for the coefficients \(c_i\),

\[\dot{c}_k = - i \frac{E_k}{\hbar} c_k \ ,\]

whose solution is \(c_k(t) = c_{k,0} \exp \left( - i \frac{E_k}{\hbar} t \right)\). The evolution of the state is

\[| \Psi_t \rangle = | \Psi_k \rangle c_{k,0} \exp\left( - i \frac{E_k}{\hbar} t \right) = | \Psi_k \rangle \langle \Psi_k | \Psi_0 \rangle \exp\left( - i \frac{E_k}{\hbar} t \right)\]

Properties. If an isolated system starts in an eigen-state of the Hamiltonian operator, i.e. \(| \Psi_0 \rangle = | \Psi_a \rangle\), \(|c_{k,0}|^2 = \delta_{ka}\), then the evolution of the system reads

\[| \Psi_t \rangle = | \Psi_a \rangle \exp\left( - i \frac{E_a}{\hbar} t \right) \ ,\]

i.e. the probability of measuring the \(k^{th}\) energy is

\[p(E = E_k) = | \langle \Psi_k | \Psi_t \rangle |^2 = \left| \underbrace{\langle \Psi_k | \Psi_a \rangle}_{\delta_{ka}} \exp\left(-i \frac{E_a}{\hbar} t\right) \right|^2 = \delta_{ka} \ ,\]

i.e. there’s probability of finding it in the same state as the initial state \(a\) is \(p(E = E_a) = 1\), while the probability of finding in any other state is zero, \(p(E = E_k) = 0\), \(k \ne a\).

15.2.3. Space and momentum operators

Position operator

Definition

\[\hat{\mathbf{r}} | \mathbf{r} \rangle = \mathbf{r} | \mathbf{r} \rangle\]

Wave function in space basis

\[\Psi(\mathbf{r}, t) := \langle \mathbf{r} | \Psi_t \rangle \ .\]

Orthonality, and identity operator

\[\begin{split}\begin{aligned} 1 & = \int_{\mathbf{r} \in \Omega} \Psi^*(\mathbf{r},t) \Psi(\mathbf{r},t) d \mathbf{r} = \\ & = \langle \Psi_t | \, \underbrace{\int_{\mathbf{r} \in \Omega} | \mathbf{r} \rangle \langle \mathbf{r} | d \mathbf{r}}_{= \hat{\mathbf{1}} } \, | \Psi_t \rangle = \\ & = \langle \Psi_t | \Psi_t \rangle \ . \end{aligned}\end{split}\]

So that the identity operator in space basis reads \(\hat{\mathbf{1}} = \int_{\mathbf{r} \in \Omega} | \mathbf{r} \rangle \langle \mathbf{r} | d \mathbf{r}\). A wave function can be thus written as

(15.3)\[| \Psi \rangle = \int_{\mathbf{r}' \in \Omega} | \mathbf{r}' \rangle \langle \mathbf{r}' | \, d \mathbf{r}' \, | \Psi \rangle \ .\]

Expansion in space basis

\[\begin{aligned} \langle \mathbf{r} | \Psi \rangle = \Psi(\mathbf{r},t) & = \int_{\mathbf{r}' \in \Omega} \delta(\mathbf{r} - \mathbf{r}') \Psi(\mathbf{r}',t) \, d \mathbf{r}' \end{aligned}\]

Comparing the last expression, with the projection of the expression (15.3) over \(\langle \mathbf{r} |\),

\[\begin{aligned} \langle \mathbf{r} | \Psi \rangle = \langle \mathbf{r} | \ , \int_{\mathbf{r}' \in \Omega} | \mathbf{r}' \rangle \langle \mathbf{r}' | \, d \mathbf{r}' \, | \Psi \rangle = \int_{\mathbf{r}' \in \Omega} \langle \mathbf{r} | \mathbf{r}' \rangle \langle \mathbf{r}' | \Psi \rangle \, d \mathbf{r}' = \int_{\mathbf{r}' \in \Omega} \langle \mathbf{r} | \mathbf{r}' \rangle \Psi(\mathbf{r'},t ) \, d \mathbf{r}' \ , \end{aligned}\]

if follows the orthogonality condition \(\langle \mathbf{r} | \mathbf{r}' \rangle = \delta(\mathbf{r} - \mathbf{r}')\).

The classical Hamiltonian of the system reads

\[H(\mathbf{r}, \mathbf{p}) = \frac{| \mathbf{p} |^2 }{2m} - \frac{q^2}{4 \pi \varepsilon | \mathbf{r} |} \ ,\]

the promotion to the Hamiltonian operator is thus

\[\hat{H} = \frac{| \hat{\mathbf{p}} |^2}{2 m } - \frac{q^2}{4 \pi \varepsilon} \hat{|\mathbf{r}|}^{-1} \ .\]

Power of an operator

Positive integer power

\[\hat{A}^n = \underbrace{\hat{A} \dots \hat{A}}_{\text{$n$ times}}\]

Let the eigenproblem of the operator be \(\hat{A} | a \rangle = a | a \rangle\), with \(a \in \mathbb{R}\). Then, the eigenproblem for the operator \(\hat{A}^n\) reads

\[\hat{A}^n | a \rangle = a^n | a \rangle \ .\]

The inverse of an operator has the eigenproblem

\[\hat{A}^{-1} | a \rangle = \frac{1}{a} | a \rangle \ ,\]

as \(| a \rangle = \hat{A} \left( \hat{A}^{-1} | a \rangle \right) = \frac{1}{a} \hat{A} | a \rangle = \frac{1}{a} \, a | a \rangle = | a \rangle\).

Vector operators

Momentum operator, \(\hat{\mathbf{p}}\), in space base \(\langle \mathbf{r} | \hat{\mathbf{p}} = -i \hbar \nabla_{\mathbf{r}} \langle \mathbf{r} |\). Using Cartesian coordinates

\[ \hat{\mathbf{e}}_a \, \langle \mathbf{r} | \hat{p}_a = - \hat{\mathbf{e}}_a \, i \hbar \partial_a \langle \mathbf{r} |\]

The operator of the square magnitude reads

\[\hat{\left( | \mathbf{p}|^2 \right)} = \hat{\left( p_x^2 + p_y^2 + p_z^2 \right)} = \hat{p_x^2} + \hat{p_y^2} + \hat{p_z^2} \ , \]

and in space coordinates

\[\langle \mathbf{r} | \hat{\left( |\mathbf{p} |^2 \right)} = - \hbar^2 \partial_{aa} \langle \mathbf{r} | = - \hbar^2 \nabla^2 \langle \mathbf{r} | \]