PHYS212 : Lectures

Next: 7.5 Bound Particles Up: 7. Quantum Mechanical Waves Previous: 7.3 Continuity and Probability

Subsections

7.4 Free Particles

7.4.1 Potential Steps

• An interface between two media in which particles have different but constant potential energies and is the quantum mechanical version of the planar interface between media we considered for acoustic (Section 4.3.2) and electromagnetic waves (Section 5.5).

• In practice, quantum potential steps are rather artificial constructions. We could present a potential ramp'' to a charged particle using a parallel plate capacitor with holes drilled in the plates allowing particles to pass through. Bringing the plates very close together would approximate a step.

• You will consider Snell's law for two dimensional reflection and refraction from such a step for homework. Here, we will continue to confine ourselves to one dimension.

• For a plane wave incident from the left with , we have
 (7.43)

with
 (7.44)

• You will show for homework that the probability current is given by
 (7.45)

• We define the reflection and transmission coefficients for a plane wave incident normally on a potential step in terms of probability currents
 (7.46)

• We can find relationships between the amplitudes , , and by matching the wavefunction and its slope at the step,
 (7.47)

which combine to eliminate either or to give
 (7.48)

• The transmission coefficient can be written in a symmetrical form
 (7.49)

which leads to the conclusion that upward and downward steps between identical and have identical transmission and reflection properties.

7.4.2 Square Barriers and Tunneling

• Consider plane waves incident on a potential barrier (or well) given by
 (7.50)

• The wave function is divided into three regions,
 (7.51)

with
 (7.52)

• The transmission and reflection coefficients of the barrier (or well) are given by Eq.'s 7.46 with the , , and of Eq. 7.51, and as with the potential step, relationships among these coefficients are found via boundary matching.

• Matching and at we find
 (7.53)

and at
 (7.54)

• These four equations cannot be solved for five unknowns (the amplitudes), but they can be solved for four amplitude ratios. We are interested in and in particular.

• With some algebraic exertion, Eq.'s 7.53 and 7.54 can be combined to eliminate the intermediate amplitude ratios and
 (7.55)

• Taking the complex square of Eq. 7.55 we have
 (7.56)

• The wavenumber is on either side of the barrier (or well), so the reflection and transmission coefficients (Eq. 7.46) are simply
 (7.57)

which together with and Eq. 7.56 gives
 (7.58)

• In terms of the energies,
 (7.59)

• Fig. 7.1 shows plots of vs. for electrons scattering from a barrier of height 1 eV and a well of depth -1 eV. Note the effect of narrowing the well.

• Note that in Fig. 7.1(a), the transmission probability is not zero for . A quantum mechanical particle can tunnel'' through barriers greater than its total energy. The sine function in Eq. 7.59 has an imaginary argument for , giving exponential instead of oscillatory behavior.

• Wells and barriers have at particular energies corresponding to the roots of the factor in Eq. 7.59,

which correspond to energies7.4
 (7.60)

Hence, we can learn about the structure of the well ( and ) from scattering measurements (transmission vs. energy).

 (a) (b)

• We also see tunneling through the barrier. That is, in Fig. 7.1(a) we have for .

7.4.3 Potential Ramps and the WKB Approximation

7.4.3.1 The Method Revisited

• The WKB method presented in Section 3.2.1 can be adapted to quantum mechanical systems. It is a suitable approach to systems, such as potential ramps, with potential energy functions which vary gradually relative to local wavelengths.

• The WKB approximation involves only the position-dependent part of the wave function. The time-inependent parts of the Scrödinger equation and the classical wave equation have the same form, and so do their solutions. Hence, the quantum mechanical version of the WKB approximation is the same as the classical version developed in Section 3.2.1.

• Plane wave WKB wavefunctions take the same form in the quantum and classical cases,
 (7.61)

• With the WKB wavefunction, Eq. 7.16 becomes
 (7.62)

• In the third term in the real part of Eq. 7.62 we recognize the local wavenumber,
 (7.63)

• In classical applications of the WKB method, the problem is essentially finding the function . In quantum mechanical applications, we have a recipe (Eq. 7.63).

• Once is defined, the real and imaginary parts of Eq. 7.62 are identical to Eq.'s 3.17 and 3.18, and we can draw the same conclusions we drew in the classical case.

• Using the explicit form of given by Eq. 7.63, and noting that we can state the WKB phase and amplitude functions in terms of either energy or momentum as
 (7.64)

and
 (7.65)

• Finally, the complete solutions take the form
 (7.66)

or alternatively
 (7.67)

• The probability density is then
 (7.68)

• The dependence of the wavefunction and probability density on the momentum indicate
• Greater momentum smaller amplitude and smaller peak
• Greater momentum larger local , shorter local

• We can state the limit of the validity of the approximation, given by Eq. 3.23 in terms of as

7.4.3.2 An Electron gun

• Linear potential ramps arise from constant, uniform force fields. The gravitational field near the surface of the earth and the electric field between two parallel charged plates are good approximations to uniform fields.

• As an example, consider the limits on the WKB approximation for electrons in a uniform electric field - an electron gun. The potential energy of an electron in an electric field oriented along is given by7.5

where is the magnitude of the field.

• The limit of validity of the WKB approximation is

which, recognizing as the worst case (shortest local wavelength), can be simplified to give
 (7.69)

• For electrons, the limit is

where is expressed in eV, which corresponds to 52 V/nm or 52 MV/mm for 10 eV electrons. This is not a difficult constraint to satisfy.

7.4.4 Gaussian Wave Packets

• We considered a classical Gaussian wave pulse in Section 2.2.5.

• In the quantum mechanical context, a traveling pulse represents a traveling particle'' with well defined position.

• Our quantum mechanical packet takes the same initial form as the classical one (Eq. 2.55 shown in Fig. 2.4),

which we can normalize

to find
 (7.70)

• The Fourier spectrum of a quantum mechanical wavefunction is called its k-space wavefunction or momentum-space wavefunction . Following the classical calculation of Section 2.2.5, we have
 (7.71)

• Eq. 7.71 describes a Gaussian k distribution centered on . Examples are shown in Fig. 2.5.

• Classical, nondispersive waves all move at the same speed, independent of wavenumber, so our classical Gaussian packet could only travel at a single, well-defined speed. However, an average of zero means zero average momentum in the quantum mechanical context. We would like a traveling packet, so we will shift our wavenumber distribution so that it is centered on a nonzero average wavenumber ,
 (7.72)

• We are now prepared to find the full time dependent wavefunction via the inverse Fourier transformation7.6
 (7.73)

• It simplifies the mathematics (but only a little) to center the integral on with the substitution , yielding
 (7.74)

• Expanding the binomials and removing constant terms from the integral, we have
 (7.75)

with
 (7.76)

where we have defined the average frequency
 (7.77)

• To simplify completing the square in the exponent in , we will further define
 (7.78)

and
 (7.79)

giving
 (7.80)

• Completing the square in the exponent, we find
 (7.81)

• With the substitution

we can write the integral

• The position space wavefunction is therefore
 (7.82)

• Eq. 7.82 is a product of a plane wave with wavenumber and a Gaussian envelope with time-dependent, complex width and amplitude. The peak of the Gaussian envelope follows the trajectory of the root of ,
 (7.83)

which moves to the right with the classical velocity of a particle of mass and momentum .

• The probability density
 (7.84)

requires some manipulation. Noting that

we find
 (7.85)

• Finally, it is convenient to write the probability in terms of its time dependent width
 (7.86)

as
 (7.87)

Fig. 7.2 shows the time evolution of a 10 eV electron with  nm.

 Copyright © 2003-2009, Lewis A. Riley Updated Wed Jan 18 09:51:28 2006