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A Dirichlet boundary condition at a hard endpoint and a single smooth classical turning point give the leading WKB quantization condition ${\int }_0^{x_{\mathrm{t}}}k(x)dx=(n+3/4)\pi$, with $n=0,1,\dots$. The phase differs from the two-soft-turning-point condition because the hard wall imposes an exact wavefunction zero rather than an Airy function connection phase. The WKB validity condition is still checked in the interior.

For the normalized ground state of a reduced quantum harmonic oscillator, ${\varphi }_0(q)={\pi }^{-1/4}{e}^{-q^2/2}$ and $E_0=1/2$. Its classical turning points are $q=\pm 1$. The Born rule assigns probability $P=2{\pi }^{-1/2}{\int }_1^{\infty }{e}^{-q^2}dq=\mathrm{erfc}(1)\simeq 0.157299$ to the classically forbidden region. This is a stationary probability of detection outside the classical interval. It is not a transmitted flux or an escape rate: the real stationary wavefunction has zero probability current.

For a one-dimensional quantum harmonic oscillator, let $b=\sqrt{\hslash /(m\omega )}$ and $\xi =x/b$. The stationary Schrödinger equation gives the displayed curvature relation. Inflections occur at classical turning points $\xi =\pm \sqrt{2n+1}$ and at simple wavefunction wavefunction nodes, provided the curvature changes sign. Thus the first excited state has an inflection at its central wavefunction node as well as at both turning points. Restricting attention to where $V=E$ misses wavefunction node inflections.

Adjacent nodes of a wavefunction are separated by roughly $a/n$, so a locally sinusoidal wavefunction has typical wavevector $k\sim \pi n/a$. The kinetic-energy estimate from wavefunction nodes is thereforeEquivalently, $⟨K⟩={\hslash }^2\int |{\psi }^{\prime }{|}^{2}dx/(2m\int |\psi {|}^{2}dx)$. The robust scaling is ${\hslash }^2{n}^2/(m{a}^2)$; the precise coefficient depends on how the extent is defined and how the oscillations vary near classical turning points. For few nodes, use an order-one offset to $n$: the nodeless ground state still has confinement energy bound from position support of order ${\hslash }^2/(m{a}^2)$, rather than zero.

The normalized quantum harmonic oscillator ground state has $|A{|}^2=\sqrt{m\omega /(\pi \hslash )}$ and energy $E_0=\hslash \omega /2$. Its classical turning points are $x=\pm \sqrt{\hslash /(m\omega )}$. With $y=x\sqrt{m\omega /\hslash }$, the ground-state oscillator probability beyond the turning points isThe stated rounded error function gives $0.16$; the more precise Gaussian integral is $0.15730$.

Quantum tunnelling is penetration of, or transmission through, a finite potential barrier into a classically forbidden region, where the total energy is less than the potential energy. The wavefunction generally remains nonzero there, so the Born rule assigns a nonzero probability of detection. This does not require an increase in total energy; it is the classical relation between a definite position and a definite kinetic energy that cannot be imposed on the quantum state. For a bound quantum harmonic oscillator, the relevant quantity is the probability of finding the particle beyond its classical turning points; it is not an escaping transmitted beam.

For the reduced quantum harmonic oscillator, $V(q)=q^2/2$ and the ground-state energy is $E_0=1/2$. Thus the classical turning points satisfy $q^2/2=1/2$, giving $q=\pm 1$. Normalizing the wavefunction gives ${\varphi }_0(q)={\pi }^{-1/4}{e}^{-q^2/2}$, because ${\int }_{-\infty }^{\infty }{e}^{-q^2}dq=\sqrt{\pi }$. The Born rule therefore gives the harmonic-oscillator forbidden-region probabilityThe factor two includes both tails. Using the exact complementary error function gives $\mathrm{erfc}(1)=0.1572992$. A stationary real wavefunction has zero probability current, so this result describes forbidden-region occupancy, not continuous loss of the bound particle.

Beyond the outer classical turning point, let $\chi (x)=\sqrt{2m[V(x)-E]}$ and $U(x)={\hslash }^{-1}{\int }_{x_2}^x\chi (y)dy$. The bound state condition removes the growing branch at positive infinity, leavingHere $F$ is an arbitrary common amplitude pending the matching and wavefunction normalization. For the confining potential energy in the native diagram, the decaying branch gives the required exterior square-integrable function.

In the central classically forbidden region, define the positive decay momentum $\chi (x)=\sqrt{2m[V(x)-E]}$ and the dimensionless action integral $K(x)={\hslash }^{-1}{\int }_x^{x_1}\chi (y)dy$. The general leading WKB approximation isBoth coefficients are allowed: this interval has finite length, so the exponential that grows towards the barrier centre cannot be discarded by a condition at infinity. The WKB validity condition holds away from the classical turning point $x_1$; a local Airy function solution supplies its continuation there.

The WKB approximation fails at a simple classical turning point $x_t$, because $p(x_t)=0$ makes its amplitude singular and violates the WKB validity condition. Locally replace $V(x)-E$ by $V^{\prime }(x_t)(x-x_t)$. For a forbidden region on the right, $V^{\prime }(x_t)>0$ and the rescaled coordinate $z=[2m{V}^{\prime }(x_t)/{\hslash }^2{]}^{1/3}(x-x_t)$ changes the time-independent Schrödinger equation into the Airy ordinary differential equation ${\psi }_{zz}-z\psi =0$. Reversing the coordinate handles a forbidden region on the left.

The two independent Airy functions have asymptotic forms, for $z,w>0$,Match these to the WKB approximation in an overlap region where the linearization of the potential energy and the large-argument asymptotic expansions both apply. If $K_t={\hslash }^{-1}|{\int }_{x_t}^x\chi dx|$ on the forbidden side and $s_t={\hslash }^{-1}|{\int }_{x_t}^{x}pdx|$ on the allowed side, the Airy turning-point connection formula isThus a decaying exponential acquires a factor two in its allowed-side amplitude, whereas a growing exponential does not. The phase convention uses positive integrals measured away from the classical turning point, so it applies at both ends of the well without an extra orientation sign. These are leading semiclassical matching rules for simple, separated classical turning points.

Use the amplitudes and positive action integrals of part (a), and define the right-well phaseMatching at the inner classical turning point gives $C=A$, $D=2B$. Matching the exterior decaying wavefunction at the outer classical turning point givesThe trigonometric identitytherefore yieldsAll three pieces on $x>0$ are now fixed up to $E$ and $F$:The Airy function descriptions replace the apparent singularities at the classical turning points. For the permitted energy eigenvalues, part (d) extends this right-hand wavefunction to the left by even or odd parity; $F$ is then determined by wavefunction normalization. At arbitrary $E$, the decaying right-hand solution instead has a unique continuation by the Schrödinger equation, which need not decay at negative infinity.

For $V(-x)=V(x)$, the parity operator $P\psi (x)=\psi (-x)$ commutes with the Hamiltonian operator. Also $P^2=I$, so its eigenvalues are $\pm 1$ and the energy eigenspaces admit a basis of definite parity. More strongly, nondegeneracy of one-dimensional bound states says that each bound state energy here is nondegenerate: the Wronskian of two solutions at the same energy is constant by the Schrödinger equation, and decay at infinity makes it zero, forcing linear dependence. Thus each bound state itself is even or odd.

An even function obeys the Neumann boundary condition ${\psi }^{\prime }(0)=0$; an odd function obeys the Dirichlet boundary condition $\psi (0)=0$. WriteSmooth reflection symmetry gives ${\chi }^{\prime }(0)=0$, while $K^{\prime }(0)=-\chi (0)/\hslash$. Hence, for the central WKB approximation from part (c),The two boundary conditions consequently give the WKB quantization condition for a symmetric double well:The full wavefunction is ${\psi }_{+}(x)={\psi }_R(|x|)$ in the even case and ${\psi }_{-}(x)=\mathrm{sgn}(x){\psi }_R(|x|)$ in the odd case, with the value at zero fixed continuously. The relevant boundary condition makes these extensions differentiable at zero. Both action integrals $\theta ,\varphi$ depend on $E$ through the integrands and the classical turning points.

Let $X=ϵx$ and suppose $k^2(X)$ stays positive and sufficiently smooth in the region considered. Write $y=\exp [{ϵ}^{-1}{S}_0(X)+S_1(X)+\cdots ]$ in ${ϵ}^2{y}_{XX}+k^2(X)y=0$. The leading and first transport equations of the WKB approximation areWith $K=\sqrt{k^2}>0$, they give $S_0^{\prime }=\pm iK$ and $S_1=-\frac{1}{2}\log K+\text{constant}$. Thus the leading WKB approximation for a slowly varying oscillator isThe amplitude is fixed by the transport equation, not just by replacing the constant wave number in a sinusoid. The local validity condition is that variation of $K$ over a wavelength is small, for example $|dK/dx|/K^2\ll 1$.

If $k^2=-{\kappa }^2<0$, the same calculation gives the evanescent formAt a classical turning point where $k^2$ vanishes, these amplitudes diverge and the separated WKB descriptions cease to be uniform. For a simple zero, an Airy turning-point connection formula replaces them in an $O({ϵ}^{2/3})$ interval in $X$; it connects oscillation to exponential behavior with the usual quarter-phase shift. A higher-order zero needs a different local canonical problem, so the Airy statement assumes a simple turning point.

For a smooth positive weight $r$ on the closed interval, let $T={\int }_0^{\pi }\sqrt{r(x)}dx$. The local wave number in the Sturm-Liouville problem is $\sqrt{\lambda }\sqrt{r}$, and the condition at zero selects a sine. Imposing the condition at $\pi$ gives the WKB quantization conditionFor weighted normalization ${\int }_0^{\pi }r|y_n{|}^{2}dx=1$, rapidly oscillating averaging gives $C_n\sim \sqrt{2/T}$. There is no turning-point phase correction here because $r$ stays positive at both endpoints with Dirichlet boundary conditions.

In the specified example $r(x)=(x+1{)}^{-2}$, so $T=\log (1+\pi )=L$ and ${\lambda }_n^{\mathrm{WKB}}=(n\pi /L{)}^2$. To solve exactly, set $t=\log (x+1)$ and $y=\sqrt{x+1}w(t)$. Direct differentiation reduces the Euler-Cauchy equation to $w^{\prime \prime }+(\lambda -1/4)w=0$ on $0<t<L$, with zero endpoint values. Hence the inverse-square weighted Dirichlet spectrum isValues $\lambda \le 1/4$ give no nonzero solution satisfying both endpoint conditions. The WKB eigenfunction shape after leading quantization is actually identical to this exact shape; the leading eigenvalue omits $1/4$.

Using the exact eigenvalue as the reference, its relative error isThus $n\ge 3$ is sufficient and necessary among positive integers. Referencing the approximate value instead gives $n>5L/\pi$, and the same integer answer.

Use primes for differentiation with respect to $X$. Since $X=ϵx$, substituting the amplitude-phase representation into the differential equation and separating real and imaginary parts givesThe second equation is exactly $(\sigma A^2{)}^{\prime }=0$. Thus, with a real constant $C$ and $A\ne 0$,This exact amplitude-phase equation is the Ermakov-Pinney equation for the oscillator. Conversely, any real nonvanishing $A$ satisfying it and $\sigma =C/A^2$ gives an exact complex solution. Existence is not an asymptotic assumption: choose independent real solutions $u,v$ of the original equation, form $y=u+iv$, and set $A=(u^2+v^2{)}^{1/2}$. Their nonzero constant Wronskian prevents simultaneous zeros. The phase derivative satisfies $\sigma A^2=u{v}_x-u_{x}v$, so it is constant. Complex conjugation gives a second branch, and linear combinations recover real solutions.

For the WKB approximation, work on an interval where $K=|k|>0$ is smooth and bounded away from zero. Choose the positive phase branch and an $ϵ$-independent positive $C$. The equations naturally admit even-power expansionsThere is no order-$ϵ$ phase correction in this convention. Leading order gives ${\sigma }_0=K$ and $A_0=(C/K{)}^{1/2}$. Comparing order ${ϵ}^2$ in the real equation yieldsConsequentlyThe constants absorb the normalization. The opposite phase branch changes the sign of $\sigma$ and its correction. A zero of $k$ is a classical turning point where this particular expansion fails; the exact amplitude-phase equation itself remains valid for a suitable complex solution.

For the Sturm-Liouville problem, take $ϵ={\lambda }^{-1/2}$ and use the interval coordinate as $X$, so $K(X)=X+1$ on $[0,1]$. Combining the conjugate branches to satisfy the left Dirichlet boundary condition produces a sine of the phase integral from zero. The right Dirichlet boundary condition requires an integer multiple of $\pi$. No turning-point phase shift is present. We haveThus the two-term Dirichlet WKB quantization isSolving first for the square root and then squaring givesAs an independent sign check, the Liouville transformation of a second-order equation with $t=X+X^2/2$ and $v=\sqrt{X+1}y$ gives $-v_{tt}-3v/[4(2t+1{)}^2]=\lambda v$ on an interval of length $3/2$. The mean transformed potential is $-3/16$, consistent with the constant spectral correction.

Let $K(X)=|k(X)|>0$ on a smooth interval, with $X=ϵx$. For the WKB approximation for a slowly varying oscillator, writeSubstitution into ${ϵ}^2{y}_{XX}+K(X{)}^{2}y=0$ gives, at the first two orders,Thus $S_0^{\prime }=\pm iK$ and $S_1^{\prime }=-K^{\prime }/(2K)$. The leading WKB approximation isThe $K^{-1/2}$ amplitude is the transport correction accompanying the rapid wave phase. Real solutions are equivalent real sine and cosine combinations.

For validity, $K$ must be smooth, nonzero and slowly varying compared with the local wavelength. In physical $x$ derivatives, sufficient local checks are $|K_x|/K^2\ll 1$ and $|K_{xx}|/K^3\ll 1$. Indeed each displayed branch has relative residualSmooth positive $K(X)$ bounded away from zero has these properties on fixed slow intervals. At a zero of $k$ the WKB approximation fails and a classical turning point requires a different local analysis, often an Airy turning-point connection formula. Rapid variations, coefficient singularities and excessively long accumulation of wave phase error also lie outside this leading approximation.

Between the classical turning points, $q<0$, while boundedness selects exponentially decaying WKB approximation solutions in the forbidden regions $x<a$ and $x>b$. The Airy turning-point connection formula at the left turning point matches the left decaying solution toin the allowed region. Matching the right decaying solution across $b$ givesFor these expressions to be proportional throughout $(a,b)$, their phases must be compatible. Each simple turning point contributes a phase shift $\pi /4$, giving the two-turning-point WKB quantization condition

The equation has the one-dimensional effective potential per unit massThus the conserved specific energy isThe potential has its minimum $-2mk/a$ at $z=0$ and tends to $0$ from below as $|z|\to \infty$. Therefore $-2mk/a<\mathcal{E}<0$ gives bounded oscillation between two turning points; $\mathcal{E}=0$ gives marginal escape with asymptotic speed zero; and $\mathcal{E}>0$ gives escape with nonzero asymptotic speed. The minimum itself is the equilibrium $z=0$.

The endpoints are the two classical turning points, determined by $V(x)=E$. Since the potential is even, they are symmetric. For $0\le E\le 1$, they lie in the linear part of the potential and satisfy $|x|/4=E$; for $E\ge 1$, they lie in the outer part and satisfy $\sqrt{|x|}-1=E$. HenceThe two-turning-point WKB quantization condition uses

Set $X=ϵx$. The equation becomesSince $k$ is positive for every $X\ge 0$, there is no classical turning point. The leading WKB approximation for a slowly varying oscillator isThe condition $y(0)=0$ selects a sine, and $y_x(0)=1$ fixes its coefficient. Sincethe result isThe WKB validity measureis uniformly $O(ϵ)$ and tends to zero at both ends of $X\ge 0$. With no turning point, the approximation is therefore uniformly valid on the entire stated half-line.

With $X=ϵx$, the equation isThe leading WKB approximation for a slowly varying oscillator isIt requires smooth nonzero $k$, $ϵ|k^{\prime }|/|k{|}^2\ll 1$, and distance from every classical turning point large compared with its turning-point scale.

For $k(X)=\sqrt{1+X^2}$,Since $k(0)=1$ and $k^{\prime }(0)=0$, the initial data select the cosine branch without an $O(1)$ phase correction. Hence, for $x>0$ in the WKB regime,