Quantum Approximation Techniques
Approximation techniques are methods through which approximate solutions to generally
hard to solve problems are generated systematically. Three common approximation methods
in quantum mechanics are: perturbation theory, applications of the variational principle,
and the WKB approximation. All of these techniques, in so far as they're related to
quantum mechanics, are discussed below.
Perturbation Theory
Perturbation theory is an approximation technique suited to problems with some resemblance to an easier (already solved) problem.
The technique is to Taylor expand about the known solution to the unperturbed problem in terms of the perturbation's coupling constant.
Then, for small values of this coupling constant, the series should converge rather nicely.
Motivational Example
As an explicit example of general perturbation theory, we'll solve for a root of a quintic polynomial of the form below, which we'll denote \(\psi(x)\)
\begin{align}
\psi(x)&=x^5-x-1=0\\
\end{align}
The trick then is to slip a coupling constant in front of one of the terms to make the problem immediately soluble. Here, one option is to place a
coupling constant, \(\epsilon\), in front of the first order term. Which, when this constant is turned to zero, yields a polynomial with a solution we already know.
This already solved problem achieved when the constant is set to zero is called the unperturbed problem, which we label \(\psi_{0}(x)\). Hence, we get
\begin{align}
\psi_c(x,\epsilon )&=x^5-\epsilon x-1=0\\
\\
\psi_c(x,1)&=\psi(x)\\
\psi_c(x_0,0)&=\psi_0(x_0)=0
\\
\end{align}
Where the unperturbed problem has a clear solution \(x_0=1\), since \(\psi_0(x)=x^5-1\). Now, the fundamental assumption of perturbation theory is that we can
expand the new solution to \(\psi(x)=0\) in terms of the coupling constant \(\epsilon \). This is a valid assumption in this case, though we must worry about
the convergence of our resulting expansion. Explicity, we expand \(x\) in terms of \(\epsilon\) as
\begin{align}
x(\epsilon)&=\sum_{n=0}^{\infty}c_n \epsilon ^n\\
&=x_0 + \sum_{n=1}^{\infty}c_n \epsilon ^n\\
\end{align}
which we may insert into the problem equation to achieve the series relation
\begin{align}
\psi_c(x,\epsilon)&=\left(\sum_{n=0}^{\infty}c_n \epsilon ^n\right)^5-\epsilon\left(\sum_{n=0}^{\infty}c_n \epsilon ^n\right)-1=0\\
&=\sum_{n=0}^{\infty}\begin{pmatrix}5\\ n\end{pmatrix} c_n \epsilon ^n-\epsilon\left(\sum_{n=0}^{\infty}c_n \epsilon ^n\right)-1=0\\
&=\sum_{n=0}^{\infty}\left[\begin{pmatrix}5\\ n\end{pmatrix} c_n-c_{n\small{-1}}-\delta(n)\right] \epsilon ^n=0\ .
\end{align}
Now, since Taylor series are unique, we may equate the coefficients of the powers of epsilon on both sides of the equation. Evidently, for each \(n\),
the coefficients above must vanish. Furthermore, this series just so happens to have a radius of convergence that contains our point of interest, that is
when \(\epsilon=1\). Below we solve for the first few terms.
\begin{align}
5c_1x_0^4-x_0=0& \ \ \ & \rightarrow c_1&=\frac{1}{5}\\
5c_2x_0^4+10c_1^2x_0^3-c_1=0 &&\rightarrow c_2&=-\frac{1}{125}\\
5c_3x_0^4+20c_2c_1x_0^3+10c_1^3x_0^2-c_2=0 &&\rightarrow c_3 &=-\frac{7}{625}\\
\end{align}
Summing together these first few terms, we arrive at an approximate solution to \(\psi(x,1)=0.\)
$$
x\approx 1+\frac{1}{5}-\frac{1}{125}-\frac{7}{625}=1.1808
$$
Now Wolfram|Alpha gives a numerical result of \(x\approx 1.1673\), so we're a little off, but still in the right direction.
Quantum Framework (Time Independent Perturbation)
In the quantum case, we specify a Hamiltonian \(H\) with some similarity to a simpler problem with a Hamiltonian \(H_0\),
and parametrize it such that at a certain value of the parameter \(\lambda\), the Hamiltonian simplifies to the simpler problem \(H_0\).
This simpler problem is termed the unperturbed problem, and it must be simple enough that we can solve for the form of it's eigenvalues.
The remaining portion of the original problem that deviates from the unperturbed Hamiltonian is termed the perturbation \(V\).
\begin{align}
H&=H_0+\lambda V\\
\\
H_0 \vert\psi_n^{(0)}\rangle&=E_n^{(0)}\vert\psi_n^{(0)}\rangle\\
\end{align}
We then Taylor expand the energy of the full Hamiltonian in terms of the parameter, which
is sometimes called the coupling constant, and solve for the first few terms (provided it converges).
\begin{align}
\vert \psi_n(\lambda)\rangle&=\sum_{m=0}^{\infty}\lambda^m\vert \psi^{(m)}_n\rangle\\
E_n(\lambda)&=\sum_{m=0}^{\infty}\lambda^m E_n^{(m)}
\end{align}
Now, these forms provoke a certain equivalence in coefficients of \(\lambda\), achieved
by inserting these expansions into the Schrodinger equation, \( H\vert\psi\rangle = E\vert\psi\rangle\).
\begin{align}
\Big(H_0+\lambda V\Big)\left(\sum_{m=0}^{\infty}\lambda^m
\vert\psi^{(m)}_n\rangle\right)&=\left(\sum_{m=0}^{\infty}\lambda^m E_n^{(m)}\right)\left(\sum_{m=0}^{\infty}\lambda^m \vert\psi_n^{(m)}\rangle\right)\\
\end{align}
Expanding the first few terms then,
\begin{align}
\Big(H_0+\lambda V\Big)&\left(\vert\psi^{(0)} \rangle+\lambda
\vert\psi^{(1)}\rangle+\lambda^2
\vert\psi^{(2)}\rangle+...\right)\\&=\left(
E^{(0)}+\lambda
E^{(1)}+\lambda^2
E^{(2)}+...\right)\left(
\vert\psi^{(0)}\rangle+\lambda
\vert\psi^{(1)}\rangle+\lambda^2
\vert\psi^{(2)}\rangle+...\right)\\
\end{align}
\begin{align}
H_0\vert\psi^{(0)}\rangle
+\lambda\Big(&H_0\vert\psi^{(1)}\rangle+V\vert\psi^{(0)}\rangle\Big)+\lambda^2\Big(
H_0\vert\psi^{(2)}\rangle+V\vert\psi^{(1)}\rangle\Big)+...\\=
E^{(0)}\vert\psi^{(0)}\rangle &+\lambda\Big(
E^{(1)}\vert\psi^{(0)}\rangle+E^{(0)}\vert\psi^{(1)}\rangle\Big)+\lambda^2\Big(
E^{(2)}\vert\psi^{(0)}\rangle+E^{(1)}\vert\psi^{(1)}\rangle+E^{(0)}\vert\psi^{(2)}\rangle\Big)+...\\
\end{align}
From this, we may project onto arbitrary eigenstates \(\vert\psi^{n}\rangle\) to calculate relations for the perturbed energy states.
Below, the relations for the first and second order corrections to the energy eigenvalues are given.
Non-Degenerate Case
In the time independent case with no degeneracies, we may simply equate coefficients of lambda (in the expansion above) and project onto the
eigenstates of the unperturbed Hamiltonian to get higher order energy corrections. This gives us rather simple relations (with increasing complexity for higher
orders) for the first two corrections that may be written somewhat succintly.
First Order Correction
The relation for first order perturbation theory is stated below
\begin{align}
E_n^{(1)}&=\langle \psi_n^0 \vert V\vert \psi_n^0\rangle\\
\end{align}
Which is achieved by projecting the expanded Schrodinger equation onto the unperturbed eigenstate.
Second Order Correction
The relation for second order corrections in perturbation theory is given as
\begin{align}
E_n^{(2)}&=\sum_{k\neq n}\frac{\vert \langle \psi_k^0 \vert V\vert \psi_n^0\rangle\vert^2}{E_n^{(0)}-E_K^{(0)}}\\
\end{align}
Which is again achieved by projecting the expanded Schrodinger equation onto the unperturbed eigenstate and matching up coefficients
of lambda in the second order.
Degenerate Case
In the case of degeneracy, the perturbation may 'lift' such degeneracies. That is, the states that are degenerate in the unperturbed Hamiltonian may
become non-degenerate in the case of the perturbation. Regardless of any lifting of degeneracies, a more nuanced treatment of such states is required (as is
evidenced by the denominator of the second order non-degenerate relation).
First Order Corrections
For a system with an energy eigenvalue \(E_n\) exhibiting a \(D\)-fold degeneracy, we may construct the first order correction as a matrix of components, as below.
\[
E_n^{(1)}=\begin{Bmatrix}\langle \psi_{n,1}\vert V\vert\psi_{n,1}\rangle & \cdots &\langle\psi_{n,D}\vert V\vert\psi_{n,1}\rangle \\
\vdots & \ddots &\vdots \\
\langle \psi_{n,1}\vert V\vert\psi_{n,D}\rangle &\cdots & \langle \psi_{n,D}\vert V\vert\psi_{n,D}\rangle\\ \end{Bmatrix}
\]
This matrix may be diagonalized (provided it doesn't vanish) to find the new eigenvectors of the perturbed Hamiltonian. In it's diagonlized form, the diagonal entries will give the
first order corrections to the relevant energy state of the perturbed eigenvectors (specified in an arbitrary order given their previous degeneracy). These new eigenvectors will also be
those of the higher order corrections, with which we may use the previous relations and expansion.
Quantum Framework (Time Dependent Perturbation)
Suppose now, we have some unperturbed Hamiltonian \(H_0\) that is independent of time that undergoes some perturbation \(V(t)\) coupled by \(\lambda\).
\begin{align}
H(t)&=H_0+\lambda V(t)\\
\\
H_0 \vert n\rangle&=E_n\vert n\rangle\\
\end{align}
Since Hamiltonians are Hermitian, the set of their eigenvectors form a basis for their corresponding space. Thus, we may write the state of the system
\(H(t)\) in terms of the unperturbed Hamiltonian's eigenstates, as below.
$$
\vert\psi (t)\rangle = \sum_nc_n(t)e^{-iE_nt/\hbar}\vert n\rangle
$$
where \(c(t)\) accounts for the variance in the states due to \(V(t)\) and the exponential term is a result of our expansion being in terms
of the unperturbed Hamiltonian's eigenstates. Hence, we wish to find dynamical expressions which may give insight into the behaviour of the
coefficients of the former; that is, we would like to solve for the \(c(t)\)'s.
To do this we simply insert the above expansion into the time-dependent Schrodinger equation, as below.
\begin{align}
i\hbar \vert\psi (t)\rangle &= H(t)\vert \psi(t)\rangle \\
\sum_n\Big(i\hbar\dot{c}_n(t)+E_nc_n(t)\Big)e^{-iE_nt/\hbar}\vert n\rangle & = \sum_n\Big(E_n+V(t)\Big)c_n(t)e^{-iE_nt/\hbar} \vert n\rangle\\
\end{align}
Now the contributions from the unperturbed Hamiltonian obviously cancel, leaving us with a relation between just \(\dot{c}_n, c_n\) and \(V(t)\). This we then project
onto the set of states \(\{ e^{-iE_nt/\hbar}\vert n\rangle\}\), giving a set of coupled differential equations for the \(c_n(t)\), as below.
\begin{align}
\dot{c}_n(t)& = -\frac{i}{\hbar}\sum_k \langle n\vert V(t)\vert k\rangle c_k(t)e^{-i(E_n-E_k)t/\hbar} \\
& = -\frac{i}{\hbar}\sum_k V_{nk} c_k(t)e^{-i\omega_{nk}t}\\
\\
\text{w/} &\ \ \ \quad \omega_{nk}=\frac{E_n-E_k}{\hbar},\quad \langle n\vert V(t)\vert k\rangle = V_{nk}
\end{align}
Note that the above relation for the coefficients non-trivial time
dependence is still analytic, in that we have made no limiting assumptions yet; we have simply chosen to write our state in the basis of the unperturbed
Hamiltionian and depict time dependence as a variance in the coefficients.
First Order Correction
For a time independent unperturbed Hamiltonian \(H_0\), with a potential \(V(t)\) that was zero before some time we'll take to be \(t=0\), we have a first order
correction for coefficients of the form:
\begin{align}
c_n^{(1)}(t)& = -\frac{i}{\hbar}\sum_k \int_0^t\langle n\vert V(t)\vert k \rangle\ c_k(0) \ e^{-i(E_k-E_n)t/\hbar} dt \\
\end{align}
Dyson Series
An alternative, but equivalent, approach to time dependent perturbations (and one in which we still assume an unperturbed Hamiltonian independent of time here) is that of Dyson series.
We now describe the time evolution due to the perturbative potential \(V(t),\) as an operator applied to the state at some intial time (which we'll take to
be zero). Thus, we have:
\begin{align}
\vert \psi(t)\rangle = \mathcal{U}(t)\vert \psi(0)\rangle
\end{align}
which gives us a series expansion in integrals for the time evolution operator of the form:
\begin{align}
\mathcal{U}(t)=1+\sum_{n=1}^{\infty} \left(\frac{-i}{n}\right)^n\int_0^{t}dt_1 \int_0^{t_1}dt_2...\int_{0}^{t_{n-1}}dt_n H(t_1)H(t_2)...H(t_n)
\end{align}
Note that this is equivalent to our prior treatment of time dependent perturbation theory (that in terms of coefficients of the state); where that
approach is effectively just this approach projected onto some eigenstate basis of \(H_0\).
First Order Correction
This gives us first order correction for the time evolution due to some potential turned on at \(t=0\):
\begin{align}
\vert\psi^{(1)}(t)\rangle=-\frac{i}{n}\int_0^{t} H(t)\ dt\\
\end{align}
Variational Principle
The variational principle is most often used to solve for upper bounds on ground state energies of systems defined by their Hamiltonians.
Motivation
The variational principle leverages the fact that the ground state energy of a system is the lowest
value of the Hamiltonian applied to some arbitrary state (given the arbitrary state satisfies the
boundary conditions specified in the relevant context). This can be seen in an eigenfunction
expansion of the Hamiltonian, done below.
Since Hamiltonians are Hermitian operators, their eigenvectors are guaranteed to be orthogonal and span the entire space. These eigenvectors may further be normalized,
and, as such, we get a complete and well behaved basis of states from any relevant Hamiltonian. Explicity, for any Hamiltonian, we have:
\begin{align}
H\vert\psi_n\rangle &=\lambda_n\vert\psi_n\rangle\\
\langle \psi_m\vert\psi_n\rangle &=\delta_{mn}\\
\end{align}
This allows us to write an arbitrary vector \(\vert\phi\rangle\) within this space (one satisfying the boundary conditions of the Hamiltonian, such as continuity and convergence at infinity) as
an expansion in these eigenvectors. Thus, we may write:
\begin{align}
\vert\phi\rangle & =\sum_nc_n\vert\psi_n\rangle\\
H&=\sum_n\lambda_n\vert\psi_n\rangle\langle\psi_n\vert\\
\end{align}
with H expanded in the same basis. In this form, it is clear to see that the smallest value we may attain from the expectation value of H with respect to some arbitrary state \(\vert\psi\rangle\) will be
the lowest eigenvalue of H (which is the ground state energy of the Hamiltonian). Hence we have:
\begin{align}
E_0 \leq \langle \phi \vert H\vert\phi\rangle
\end{align}
Note that the above relation assumes \(\vert\psi\rangle\) is normalized, but the formula for application (below) omits this requirement and builds in a normalization factor.
Application
As a result of the above considerations, any 'trial' wavefucntion we construct that satisfies
the relevant conditions of a specified Hamiltonian will yield an effective upper limit on the ground state energy. Hence, we may calculate
an upper bound \(\tilde{E}_0\) on the ground state energy \(E_0\) of some Hamiltonian \(H\) with an arbitrary trial wavefunction \(\vert\psi\rangle\)
via
\begin{align}
\tilde{E}_0[\psi] = \frac{\langle \psi \vert H\vert\psi\rangle}{\langle\psi\vert\psi\rangle}\geq E_0
\end{align}
where we divide by the inner product of our trial function with itself for normalization. The trial wavefunction may further be parametrized, such that
we may find a stationary value of the upper bound, with respect to the parameter, from which we may attain a tighter upper bound.
The variational principle may also be applied to solve for upper bounds on higher energy states. Though, this is only possible if we may construct a trial wavefunction
that is guaranteed to be orthogonal to the wave functions of all the lesser energies below that under consideration (so if we wanted to calculate an \(\tilde{E}_3\), for example, we would
need a trial wavefunction orthogonal to \(\vert\psi_0\rangle,\vert\psi_1\rangle\) and \(\vert\psi_2\rangle\)).
WKB Approximation (Semiclassical)
The WKB approximation is a method for finding approximate solutions to certain classes of linear differential equations. In the current context, we wish to find approximate solutions
to the one-dimensional Schrodinger equation in appropriate circumstances. To define these circumstances succintly, and give some motivation, we first introduce some notation below and
then proceed to describe the general form of the approximate solution.
This approximate solution can then be used to calculate tunneling coefficients, that is, the probability that some quanta will tunnel through some potential barrier that exceeds it's energy.
Again, we restrict ourselves to one dimension, though the WKB approximation is also applicable in three dimensions for cases with certain symmetries.
Motivational Example
WKB theory is a class of perturbative approximations that may be used to solve for approximate solutions to ordinary differential equations of the form:
\begin{align}
\epsilon^2\frac{d^2}{dx^2}y(x) +Q(x)y(x)=0
\end{align}
Note that this a particularly peculiar perturbation as, in the limit of epsilon
approaching zero, the above turns from a differential equation into a algebraic equation. However,
we assume a series solution in \(\epsilon\) of the form given below, which turns \(y''(x)\)
into something that scales with \(1/\epsilon^2\) and, in turn, avoids the aforementioned
transition.
\begin{align}
y(x)\sim e^{\frac{1}{\epsilon}\sum_n S_n(x) \epsilon^n}
\end{align}
We now insert this form of perturbative expansion into the previous differential equation. This gives us the equality below, in which we may equate the
coefficients of each order of \(\epsilon ,\) giving us a set of equations from which we may deduce arbitrary order coefficients.
\begin{align}
\epsilon^2\frac{d^2}{dx^2}e^{\frac{1}{\epsilon}\sum_n S_n(x) \epsilon^n} -Q(x)e^{\frac{1}{\epsilon}\sum_n S_n(x) \epsilon^n}&=0\\
\epsilon^2\frac{d}{dx}\left[\left( \frac{1}{\epsilon}\sum_n S_n'(x) \epsilon^n\right)e^{\frac{1}{\epsilon}\sum_n S_n(x) \epsilon^n}\right]
-Q(x)e^{\frac{1}{\epsilon}\sum_n S_n(x) \epsilon^n}&=0\\
\epsilon^2\left[ \frac{1}{\epsilon}\sum_n S_n''(x) \epsilon^n+\left(\frac{1}{\epsilon}\sum_n S_n'(x) \epsilon^n\right)^2\right]e^{\frac{1}{\epsilon}\sum_n S_n(x) \epsilon^n}
-Q(x)e^{\frac{1}{\epsilon}\sum_n S_n(x) \epsilon^n}&=0\\
\epsilon\sum_n S_n''(x) \epsilon^n+\left(\sum_n S_n'(x) \epsilon^n\right)^2-Q(x)&=0
\end{align}
The infinite series that is squared above may then be expanded into a series over a double summation, as below.
\begin{align}
\left(\sum_n S_n' \epsilon^n\right)^2&=\sum_{n=0}^{\infty}\sum_{j=0}^{n}S'_jS'_{n-j}\ \epsilon^n
\end{align}
Which gives us the following set of equations defining every coefficient of the series expansion we assumed for \(y(x)\):
\begin{align}
S_{n-1}''(x) +\sum_{j=0}^{n}S'_jS'_{n-j}-Q(x)\delta_{n}&=0
\end{align}
The first few terms of this series, explicity, are then given below. These may be used to solve for the
coefficients of our expansion, term by term.
\begin{align}
\left(S'_{0}\right)^2&=Q(x)&\quad\ &\rightarrow&\ &S_0(x)=\pm\int_0^x \sqrt{Q(x)} dx \quad \tiny{\text{(Eikonal Equation)}}\\
S_0'' +2S_0'S_1'&=0&\quad&\rightarrow&&S_1(x)=-\frac{1}{4}\ln Q(x)\\
S_1''+2S'_0S'_2+\left(S'_1\right)^2&=0&\quad &\rightarrow&&S_2(x)=\pm \int_0^x\left[\frac{Q''}{8Q^{3/2}}-\frac{5(Q')^2}{32Q^{5/2}}\right]dx\\
\quad \quad &\ \ \vdots &\quad &&&\\
\end{align}
Now, this series is singular when \(Q(x)=0\), as can be seen in the solution for \(S_1(x)\). This is essentially our criteria of applicability for
our approximate form then. That is, the above expansion for \(y(x)\) is well-defined for \(Q(x)\neq 0\).
Hence, to first order in \(\epsilon\), we have an approximate solution for the original differential equation (for regions where \(Q(x)\neq 0\)) of the following form.
\begin{align}
y^*(x)&=e^{\frac{1}{\epsilon}\left(S_0+\epsilon S_1\right)}= Q^{-\frac{1}{4}} \exp\left(\ \pm\frac{1}{\epsilon}\int\sqrt{Q}dx\right)
\end{align}
And, to reiterate, this solution fails near points where \(Q(x)\) vanishes, but we may patch together other asymptotic approximations near these points, a technique which
is discussed below.
Quantum Framework (WKB Approx.)
Beginning with the Schrodinger equation, we first rearrange to get an effective momentum.
\begin{align}
\frac{d^2}{dx^2}\vert\psi(x)\rangle = \frac{2m}{\hbar^2}(V(x)-E)\vert\psi(x)\rangle=-\left[\frac{p(x)}{\hbar}\right]^2\vert\psi(x) \rangle
\end{align}
With this effective momentum \(p(x)=\sqrt{2m(V-E)}\), we define a de Broglie wavelength for our particle.
\begin{align}
\lambda = \frac{\hbar}{p(x)}=\frac{\hbar}{\sqrt{2m(E-V(x))}}
\end{align}
Now, when this de Broglie wavelength is much smaller than a region through which E isn't close to V(x), we expect the quantum behaviour of our particle to be suppressed (this corresponds to
our neglect of a certain contribution later).
For further motivation, we consider the fact that when the potential is some constant less than the energy of the quanta, we have a simple solution to the differential equation.
Explicity, with a simple constant potential of the form \(V(x)=C\), we have the Schrodinger equation:
\begin{align}
\frac{d^2}{dx^2}\vert\psi(x)\rangle +\left[\frac{p}{\hbar}\right]^2\vert\psi(x)\rangle=0
\end{align}
with the simple solution
\begin{align}
\vert\psi(x)\rangle\vert_{\tiny{V=C}}=e^{\pm i px/\hbar}
\end{align}
So, in the case where \(V(x)\) is non-trivial and does depend on \(x\), we adopt a solution of the form given for \(\psi\) below.
\begin{align}
\vert\psi(x)\rangle\equiv e^{ i W(x)/\hbar}
\end{align}
and insert this into the Schrodinger equation to (hopefully) solve for \(W(x)\). Substituting in this form for \(\psi\) then gives
\begin{align}
\frac{d^2}{dx^2}e^{ i W(x)/\hbar} +\left[\frac{p(x)}{\hbar}\right]^2e^{ i W(x)/\hbar}&=0\\
i\hbar\frac{d^2W(x)}{dx^2}-\left(\frac{dW(x)}{dx}\right)^2+p(x)^2&=0\\
\end{align}
which is a non-linear differential equation, completely equivalent to the Schrodinger equation (that is, we have made no approximation in the above
equation; we have only chosen to write \(\psi\) in an exponential form). We further expand \(W(x)\) in terms of \(\hbar\) as a power series:
\begin{align}
W(x) = \sum_{n=0}^{\infty}w_n\hbar^n
\end{align}
The approximation we now make is that of the WKB method; we take the leading order
derivative's (which is coupled by \(\hbar\)) contribution to be negligible compared to the other derivative. Thus, if we want our approximation to be
reasonably valid, we should only consider potentials that 'slowly vary' such that this is true. Explicity, this
leaves us with an approximate solution (in the region where this holds, and to second order in \(\hbar\)) of the form:
\begin{align}
W(x) \approx \frac{1}{\sqrt{p(x)}}\exp \left( \pm\frac{i}{\hbar}\int_0^t p(x)dx\right)
\end{align}
with the \(\pm\) depending on the whether the region has energy greater than the potential or less than the potential (respective vertically).
Below we address regions in which the potential energy is equal to the energy of the system, as the above approximation is clearly singular
in such regions.
Turning Points
Since our expansion in \(\hbar\) is only valid in regions where \(p(x)\) is non-zero (that is, when the system's energy nears the value of the potential energy), we
must approach such points of singularity in a different manner. In turn, we approximate a solution valid near the singular points of \(p(x)=0\), and then stitch it to
our approximation in \(\hbar\) by imposing some boundary conditions. Since \(p^2(x)=0\) in these regions, we may expand in terms of \(x\) itself and truncate the power series
again to get an approximate solution. Note that we choose \(p^2\) because it clearly satisfies the same property (that of vanishing at some point) and simplifies our approach.
We assume that \(p^2(x)\) vanishes at \(x=0\) (at slight loss of generality, though the below discussion can be adequatley transformed to an arbitrary turning point \(x_0\) by replacing the power
series in terms of \(x-x_0\)). Explicitly, we may expand \(p(x)\) in terms of \(x\) as
\begin{align}
\frac{-p^2(x)}{\hbar^2} = Q(x) = \sum_{n=1}^{\infty}q_n x ^n
\end{align}
In which case, the Schrodinger equation takes the simple form:
\begin{align}
\frac{d^2}{dx^2}\vert\psi(x)\rangle&=\left(\sum_{n=1}^{\infty}q_nx ^n\right)\vert\psi(x)\rangle\\
&
\end{align}
We then truncate the series to first order (assuming the expansion has a non-vanishing linear term in \(x\)), as an approximation near the point. This gives the simplified, approximate, differential
equation relevant for points where \(q(x)=p^2(x)\) vanishes:
\begin{align}
\frac{d^2}{dx^2}\vert\psi(x)\rangle&\approx q_1x \vert\psi(x)\rangle\\
\end{align}
Which can clearly be recognized as a scaled Airy Equation, with already known solutions. These solutions are
known as the Airy functions and give us an approximate solution of \(\vert\psi(x)\rangle\):
\begin{align}
\vert\psi(x)\rangle \approx C_a Ai(q_1^{1/3}x)+ C_b Bi(q_1^{1/3}x)
\end{align}
We then may stitch this approximate solution to our previous approximate solution by equating them along some boundary a small distance from the point of singularity.
Resources
Perturbation Theory
For further resources related to perturbation theory's application in quantum mechanics, see the Wikipedia article:
Perturbation theory (quantum mechanics).
Other resources include a nice introduction in chapter 5 of Sakurai and Napolitano's Modern Quantum Mechanics, and chapter 6 of David Griffith's
Introduction to Quantum Mechanics.
Variational Principle
Applications of the variational principle are covered in chapter 7 of David Griffith's
Introduction to Quantum Mechanics, section 5.4 of Sakurai and Napolitano's Modern Quantum Mechanics, and the Wikipedia article:
Variational Method (quantum mechanics).
WKB Approximation
For a good introduction to the WKB approximation in quantum mechanics, see this mini-lecture on Youtube by Barton Zwiebach. The WKB
approximation is also covered in chapter 8 of David Griffith's
Introduction to Quantum Mechanics, the Wikipedia article:
WKB Approximation, and briefly in Section 2.5.4 of Sakurai and Napolitano's Modern Quantum Mechanics.
The WKB
approach, in it's general form, is also covered in great detail in Chapter 10 of Carl Bender and Steven Orszag's Advanced Mathematical Methods for Scientists and Engineers, as well
as in lectures 12 and 13 of Carl Bender's Mathematical Physics lectures which can be found here.
