Schrödinger wave equation and Fourier transform in 1 dimension

The mathematical treatment of this subject is far from rigorous: it is intended to get a "physical touch" of the matter. This stuff shall be treated more formally in other courses

General concepts

1. Schrödinger's wave equation
   iℏ ∂/_∂t ψ = H ψ
   is a linear differential equation for the unknown function ψ(x,t).
2. The "total energy" (or Hamiltonian) operator H = T + V = p²/2m + V(x) is a differential operator, acting on ψ(x,t). The kinetic term is -ℏ²/2m times the spacial Laplace operator (as p= -iℏ∇_x). The V operator acts on ψ to give the product function V(x)·ψ(x,t).
3. The conventional interpretation of ψ is statistical: |ψ(x,t)|² dx = P(x,t) dx = probability of finding the quantum particle between x and x+dx at time t.
4. Solutions of
   H ψ = E ψ
   (where E is a number, independent of x and t) are called eigenfunctions of H. The eigenvalue E is usually called energy.
5. The time evolution of any eigenfunction of H can be computed once and for all. For the time dependence of ψ one finds that ψ(x,t_fin) = ψ(x,t_ini) ·exp[-i E (t_fin - t_ini)/ℏ] Thus, the evolved state is simply multiplied by a simple phase factor. [EXERCISE: verify this result by solving in detail the corresponding first-order differential equation.]
   In particular, the square modulus P(x,t_fin) = P(x,t_ini), as the phase factor is irrelevant. Thus, the probability distribution of an eigenfunction is independent of time (that's why eigenfunctions are also called "stationary").
   The only effect of passing time is to multiply an eigenfunction by a phase factor. Specifically, the phase is -E t/ℏ, thus it is directly proportional to the energy of the state and the elapsed time.
6. The natural space where to look for eigenfunctions of the H Schrödinger operator is the space of complex functions which are squared-norm integrable on the domain where the quantum particle is allowed to move. Consider for example particles moving on the whole straight line represented by the set of real numbers R. Let's call L²(R) the set of all complex functions f defined on R, such that ∫_-∞^∞|f(x)|²dx exists and is a (finite real positive) number. [this definition is not quite precise mathematically!]
7. EXERCISE: give 3 examples of functions belonging to L²(R), and 3 examples of complex functions defined on R but not belonging to L²(R).
8. EXERCISE: verify that L²(R) is a vector space (f₁+f₂ and also a·f₁ belong to L²(R) if f₁ and f₂ do, and if a is any given complex number).
9. In this space of functions one can define an Hermitian product <f|g> = ∫_-∞^∞f(x)^*·g(x)dx. The norm associated to this inner product is ||f||=(<f|f>)^1/2 = (∫_-∞^∞|f(x)|²dx)^1/2
10. If one thinks intuitively of two functions f and g as two vectors with infinitely many components f(x), then the definition of the inner product corresponds to that ∑_iv_i^*w_i between complex vectors in the n-dimensional vector space Cⁿ. In the same intuitive fashion one can think of <f|g>/(||f||·||g||) as the cosine of the angle formed by these two vectors. For example, two functions f and g such as <f|g>=0 are said to be orthogonal.
11. A function C from L²(R) to L²(R) is often called operator. An operator C such that C(f₁+f₂) = C f₁ + C f₂ and C(a·f₁) = a·C f₁, for any... is called a linear operator.
12. Most (or rather, all) physical observable quantities are represented by suitable linear operators. Example: the energy operator H introduced in the first two points above.

The Fourier Transform (1D)

1. A very useful linear operator on L²(R) is called Fourier transform (FT). We indicate it with F. Like any operator, the action of F on a function f gives another function F[f], sometimes indicated also as Ff. Given any function f(x) in L²(R), the FT acts as follows: F[f](k) = ∫_-∞^∞exp(-ikx) f(x) dx/(2 π)^1/2.
2. The inverse FT F^-1 is given by: F^-1[g](y) = ∫_-∞^∞exp(ikx) g(k) dk/(2 π)^1/2
3. EXERCISE: verify that both F and F^-1 are linear operators.
4. EXERCISE: verify that F^-1 is really the inverse operator of F. Hint: take any function f and apply sequentially the two transforms: F^-1[F[f]](y) and show that the resulting function equals f(y). You will need the following identity: ∫_-∞^∞exp(ikx) = 2 π δ(x), where the δ(x) distribution is defined by ∫_-∞^∞ δ(x)g(x)dx=g(0) for any sufficiently smooth function g.
5. Another important property of F and F^-1 is unitarity: <Ff|Fg> = <f|g> for any f and g in L²(R). It means that function norms and angles are preserved by this transformation. Note that often different definition of the FT are used, putting two (2 π)^-1/2 factors into the definition of F or of F^-1, instead of putting one into each as we do here. In that convention, the FTs are not unitary operators!
6. What makes the FT useful for the study of any wave phenomena, and for the Schrödinger wave equation in particular, is the fact that it maps the derivative operator into a multiplication operator: F[-i ∂/_∂x f(x)](k) = ∫_-∞^∞ exp(-ikx) (-i ∂/_∂x f(x)) dx/(2 π)^1/2 = i (-i·k) ∫_-∞^∞ exp(-ikx) f(x) dx/(2 π)^1/2 = k F[f](k) [EXERCISE: verify this result by carrying out the partial integration, in the assumption of f(x) vanishing at infinity].
7. A differential operator for f(x) becomes a simple multiplication operator its FT for F[f](k). This can be an extremely powerful tools to solve problems such as the free particle, where the Hamiltonian operator is made exclusively by differential operators.
8. In practice, as F is invertible, ψ(x) and its FT F[ ψ](k) carry the same amount of information, encoded differently. The "real-space" x description and the "Fourier-space" k description are equivalent. It is usually a matter of convenience to choose either representation.
9. Exactly the same mathematics, replacing x with t and k with ω, leads to the time-frequency Fourier transform. Our ears-brain device performs this task all the times it converts the air-pressure time series into "sounds".

Exercise

A wave packet representing a quantum particle allowed to move along a straight line is represented in Fourier space by

1. Check that this state is normalized ||φ||=1.
2. Compute the real-space representation of this wave packet ψ(x)= F^-1[φ](x).
   RESULT: (4 π · Δ)^-1/2·exp(iqx) ∫_-Δ^Δdk' exp[i(k'x-χ(k'+q))], where q=(k₁+k₂)/2 and Δ=(k₂-k₁)/2.
3. Taylor-expand about k=q the phase function χ(k'+q) = χ(q) + a·k' + O(k'²). Neglect higher-than-first order terms, and assume χ(q)=0 (it would yield a trivial constant phase factor in ψ(x)). Compute explicitly ψ(x) in this approximation. Draw a plot of P(x)=|| ψ(x)||².
   RESULT: ψ(x)=(π · Δ)^-1/2·exp[iq(x-a)] sin[ Δ(x-a)]/(x-a).
4. Evaluate the average momentum <p> of this state, both in the k representation and in the x representation, and verify that they coincide.
   RESULT: <p> = ℏq.
5. Evaluate the average kinetic energy <p²/(2m)> of this state, using the representation that you find most convenient. Compare this result with <p> of the previous question.
   RESULT: ℏ²/(2m) × (q²+ Δ²/3).
6. Evaluate the average position <x> of this state, in the most convenient representation. Meditate on the different role taken by a in the k and x representation.
   RESULT: Trivially, <x> = a.
7. Assume now that χ(k)=ω(k)·t represents the time evolution of the several k-eigenfunction of the free-particle Hamiltonian mixed in the wave packet ψ(x) by the F operator. Recalling the definition of point 3: a=^d/_dk( χ)(q), compute the velocity of displacement of the center <x> of the wave packet along the x line. Consider the two cases of a) photons ω(k)=c|k|, and b) Schrödinger material waves ω(k)= ℏ²k²/(2m).
   RESULT: a) c. b) p/m. Remark: this result illustrates the general fact that the packet velocity (the so called group velocity), coinciding with the classic velocity, equals d ω/dk computed at the average k of the packet, q.
8. Evaluate the phase velocity of each k component of the wave packet, i.e. the velocity at which each iso-|ψ|² point translates in space. Compare the Maxwell and Schrödinger waves as above.
   RESULT: In general: ω/k. Therefore a) =c; b) =p/(2m). Remark: this result illustrates that the Schrödinger wave packets move twice as fast as the underlying k-waves (verify this with the applet putting V=0: the Re( ψ) & Im(ψ) wavefronts "lag behind" the | ψ|² packet maximum). See also these clear animations to visualize group velocity and phase velocity in a k² dispersion. Carrying over these observation from material waves to electromagnetic waves, does this suggest a trick to attempt to produce light packets moving faster than c?
9. Use this exercise to explain why free wave packets disperse, i.e. change shape in time, getting generally broader (see applet, putting V=0). Under which conditions doesn't this happen?
10. Assume a smooth (but not just linear) dispersion ω(k). Take a narrow-in-k wave packet of width Δ and a broader-in-k wave packet of width 10Δ: which one of them would generally change its shape faster in time?