Square Potential Barrier

Consider a particle of mass $m$ and energy $E>0$ interacting with the simple, one-dimensional, potential barrier

$$U(x) = \left\{\begin{array}{rcl} V&\mbox{\hspace{1cm}}&\mbox{for $0\leq x\leq a$}\\ [0.5ex] 0&&\mbox{otherwise} \end{array}\right.,$$ (11.93)

where $V>0$. In the regions to the left and to the right of the barrier, the stationary wavefunction, $\psi(x)$, satisfies

$$\frac{d^{\,2} \psi}{d x^{\,2}} = - k^{\,2}\,\psi,$$ (11.94)

where

$$k = \sqrt{\frac{2\,m\,E}{\hbar^{\,2}}}.$$ (11.95)

Let us adopt the following solution of the previous equation to the left of the barrier (i.e., $x<0$):

$$\psi(x) = {\rm e}^{\,{\rm i}\,k\,x} + R\,{\rm e}^{-{\rm i}\,k\,x}.$$ (11.96)

This solution consists of a plane wave of unit amplitude traveling to the right [because the full wavefunction is multiplied by a factor $\exp(-{\rm i}\,E\,t/\hbar$)], and a plane wave of complex amplitude $R$ traveling to the left. We interpret the first plane wave as an incident particle, and the second as a particle reflected by the potential barrier. Hence, $\vert R\vert^{\,2}$ is the probability of reflection. (See Sections 6.7 and 11.6.)

Let us adopt the following solution to Equation (11.94) to the right of the barrier (i.e. $x>a$):

$$\psi(x) = T\,{\rm e}^{\,{\rm i}\,k\,x}.$$ (11.97)

This solution consists of a plane wave of complex amplitude $T$ traveling to the right. We interpret the plane wave as a particle transmitted through the barrier. Hence, $\vert T\vert^{\,2}$ is the probability of transmission.

Let us, first of all, consider the situation in which $E>V$. In this case, according to classical mechanics, the particle slows down as it passes through the barrier, but is otherwise unaffected. In other words, the classical probability of reflection is zero, and the classical probability of transmission is unity. However, this is not necessarily the case in wave mechanics. In fact, inside the barrier (i.e., $0\leq x\leq a$), $\psi(x)$ satisfies

$$\frac{d^{\,2}\psi}{dx^{\,2}} =-q^{\,2}\,\psi,$$ (11.98)

where

$$q = \sqrt{\frac{2\,m\,(E-V)}{\hbar^{\,2}}}.$$ (11.99)

The general solution to Equation (11.98) takes the form

$$\psi(x) = A\,{\rm e}^{\,{\rm i}\,q\,x}+B\,{\rm e}^{-{\rm i}\,q\,x}.$$ (11.100)

Continuity of $\psi$ and $d\psi/d x$ at the left edge of the barrier (i.e., $x=0$) yields

$$1 + R = A+B,$$ (11.101)

$$k\,(1-R) = q\,(A-B).$$ (11.102)

Likewise, continuity of $\psi$ and $d\psi/d x$ at the right edge of the barrier (i.e., $x=a$) gives

$$A\, {\rm e}^{\,{\rm i}\,q\,a}+ B \,{\rm e}^{-{\rm i}\,q\,a} = T\,{\rm e}^{\,{\rm i}\,k\,a},$$ (11.103)

$$q\left(A\,{\rm e}^{\,{\rm i}\,q\,a} -B \,{\rm e}^{-{\rm i}\,q\,a}\right) =k\,T\,{\rm e}^{\,{\rm i}\,k\,a}.$$ (11.104)

After considerable algebra, the previous four equations yield

$$\vert T\vert^{\,2} = 1-\vert R\vert^{\,2}= \frac{4\,k^{\,2}\,q^{\,2}}{4\,k^{\,2}\,q^{\,2} + (k^{\,2}-q^{\,2})^{\,2}\,\sin^2(q\,a)}.$$ (11.105)

The fact that $\vert R\vert^{\,2}+\vert T\vert^{\,2}=1$ ensures that the probabilities of reflection and transmission sum to unity, as must be the case, because reflection and transmission are the only possible outcomes for a particle incident on the barrier.

Figure: 11.8 Transmission (solid curve) and reflection (dashed curve) probabilities for a square potential barrier of width $a=1.25\,\lambda$, where $\lambda$ is the free-space de Broglie wavelength, as a function of the ratio of the height of the barrier, $V$, to the energy, $E$, of the incident particle.

Figure 11.9: Transmission (solid curve) and reflection (dashed curve) probabilities for a particle of energy $E$ incident on a square potential barrier of height $V= 0.75\,E$ as a function of the ratio of the width of the barrier, $a$, to the free-space de Broglie wavelength, $\lambda$.

The reflection and transmission probabilities obtained from Equation (11.105) are plotted in Figures 11.8 and 11.9. It can be seen, from Figure 11.8, that the classical result, $\vert R\vert^{\,2}=0$ and $\vert T\vert^{\,2}=1$, is obtained in the limit where the height of the barrier is relatively small (i.e., $V\ll E$). However, if $V$ is of order $E$ then there is a substantial probability that the incident particle will be reflected by the barrier. According to classical physics, reflection is impossible when $V < E$.

It can also be seen, from Figure 11.9, that at certain barrier widths the probability of reflection goes to zero. It turns out that this is true irrespective of the energy of the incident particle. It is evident, from Equation (11.105), that these special barrier widths correspond to

$$q\,a = n\,\pi,$$ (11.106)

where $n=1,2,3,\cdots$. In other words, the special barrier widths are integer multiples of half the de Broglie wavelength of the particle inside the barrier. There is no reflection at the special barrier widths because, at these widths, the backward traveling wave reflected from the left edge of the barrier interferes destructively with the similar wave reflected from the right edge of the barrier to give zero net reflected wave. (See Section 6.7.)

Let us now consider the situation in which $E< V$. In this case, according to classical mechanics, the particle is unable to penetrate the barrier, so the coefficient of reflection is unity, and the coefficient of transmission zero. However, this is not necessarily the case in wave mechanics. In fact, inside the barrier (i.e., $0\leq x\leq a$), $\psi(x)$ satisfies

$$\frac{d^{\,2} \psi}{d x^{\,2}} = q^{\,2}\,\psi,$$ (11.107)

where

$$q = \sqrt{\frac{2\,m\,(V-E)}{\hbar^{\,2}}}.$$ (11.108)

The general solution to Equation (11.107) takes the form

$$\psi(x) = A\,{\rm e}^{\,q\,x} + B\,{\rm e}^{-q\,x}.$$ (11.109)

Continuity of $\psi$ and $d\psi/d x$ at the left edge of the barrier (i.e., $x=0$) yields

$$1 + R = A+B,$$ (11.110)

$${\rm i}\,k\,(1-R) = q\,(A-B).$$ (11.111)

Likewise, continuity of $\psi$ and $d\psi/d x$ at the right edge of the barrier (i.e., $x=a$) gives

$$A\, {\rm e}^{\,q\,a}+ B \,{\rm e}^{-q\,a} = T\,{\rm e}^{\,{\rm i}\,k\,a},$$ (11.112)

$$q\left(A\, {\rm e}^{\,q\,a}-B \,{\rm e}^{-q\,a}\right) = {\rm i}\,k\,T\,{\rm e}^{\,{\rm i}\,k\,a}.$$ (11.113)

After considerable algebra, the preceding four equations yield

$$\vert T\vert^{\,2} = 1-\vert R\vert^{\,2}= \frac{4\,k^{\,2}\,q^{\,2}}{4\,k^{\,2}\,q^{\,2} + (k^{\,2}+q^{\,2})^{\,2}\,\sinh^2(q\,a)}.$$ (11.114)

The fact that $\vert R\vert^{\,2}+\vert T\vert^{\,2}=1$ again ensures that the probabilities of reflection and transmission sum to unity, as must be the case, because reflection and transmission are the only possible outcomes for a particle incident on the barrier.

Figure: 11.10 Transmission (solid curve) and reflection (dashed curve) probabilities for a square potential barrier of width $a=0.5\,\lambda$, where $\lambda$ is the free-space de Broglie wavelength, as a function of the ratio of the energy, $E$, of the incoming particle to the height, $V$, of the barrier.

Figure 11.11: Transmission (solid curve) and reflection (dashed curve) probabilities for a particle of energy $E$ incident on a square potential barrier of height $V = (4/3)\,E$ as a function of the ratio of the width of the barrier, $a$, to the free-space de Broglie wavelength, $\lambda$.

The reflection and transmission probabilities obtained from Equation (11.114) are plotted in Figures 11.10 and 11.11. It can be seen, from these two figures, that the classical result, $\vert R\vert^{\,2}=1$ and $\vert T\vert^{\,2}=0$, is obtained for relatively thin barriers (i.e., $q\,a\sim 1$) in the limit where the height of the barrier is relatively large (i.e., $V\gg E$). However, if $V$ is of order $E$ then there is a substantial probability that the incident particle will be transmitted by the barrier. According to classical physics, transmission is impossible when $V > E$.

It can also be seen, from Figure 11.11, that the transmission probability decays exponentially as the width of the barrier increases. Nevertheless, even for very wide barriers (i.e., $q\,a\gg 1$), there is a small but finite probability that a particle incident on the barrier will be transmitted. This phenomenon, which is inexplicable within the context of classical physics, is called tunneling. For the case of a very high barrier, such that $V\gg E$, the tunneling probability reduces to

$$\vert T\vert^{\,2}\simeq \frac{4\,E}{V}\,{\rm e}^{-2\,a/\lambda},$$ (11.115)

where $\lambda = (\hbar^{\,2}/2\,m\,V)^{1/2}$ is the de Broglie wavelength inside the barrier. Here, it is assumed that $a\gg\lambda$. Thus, even in the limit that the barrier is very high, there is an exponentially small, but nevertheless non-zero, tunneling probability. Quantum mechanical tunneling plays an important role in the physics of electron field emission and $\alpha$-decay (Park 1974). (See Sections 11.14 and 11.15.)