Visualizing the Schrodinger equation

The Schrodinger equation describes how a physical system changes over time. But what does it mean intuitively?^[1]

Imagine a particle moving freely through space. There are no forces acting on the particle so it travels in a straight line along the x-axis.

In Classical Mechanics, if the current state of the particle is known (such as its position and momentum) then its future state can be predicted according to classical laws.

In Quantum Mechanics, the state of the particle is represented by a wave function^[2] that has a complex value and is denoted by the Greek letter Ψ (psi). The state of the particle can be prepared so that its wave function is initially known and it will then evolve in time according to the Schrodinger equation. Further mathematical operations can be performed on the wave function to determine the position, momentum or energy of the particle.

The time evolution can be elegantly expressed as Ψ(t) = U(t)Ψ(t=0) where U(t) is a unitary operator that propagates the wave function from its initial configuration at time t=0 to its final configuration at time t. U(t) = e^-iEt/ħ which is an exponential formula that represents a rotation (or phase change) on the complex plane of Et/ħ radians, where E is the total energy of the system and ħ is the reduced Planck constant. The wave function (at time t=0) can be visualized as a clock hand that rotates to a new position when the operator U(t) is applied to it. The greater the energy, the more rotations per second.

So we know U(t) and how to calculate the future wave function Ψ(t). All we need is the wave function at time t=0 to plug into the equation. The simplest wave function is a plane wave that curls in a uniform spiral around the x-axis (see Figure 1 below).

Figure 1 - Complex plane wave

A plane wave has the general formula Ae^ipx/ħ where A is the wave amplitude, p is the momentum and x is the position on the x-axis. Note that, like the time evolution operator, it also has an exponential representing rotation on the complex plane. However, in this case, the rotation is across space rather than over time. The greater the momentum, the more rotations per meter.

So our initial wave function is Ψ(x,t=0) = Ae^ipx/ħ. Therefore our wave function at time t is Ψ(x,t) = e^-iEt/ħAe^ipx/ħ, which can be expressed more simply as:

Ψ(x,t) = Ae^{i(px - Et)/ħ} 

where the energy is proportional to the square of the momentum (E=p²/2m) and position and momentum are related via the canonical commutation relation xp - px = iħ.^[3]

A way to visualize this equation is to imagine the entire wave in Figure 1 to be dynamically rotating as time progresses. It will appear to be travelling along the x-axis in a periodic manner. The greater the momentum (and therefore energy), the tighter the spiral and the faster it will be spinning.

Let's suppose that we've prepared our particle to have a precise momentum (for example, 40 kgm/s where our particle weighs 5kg and has an energy of 160 joules, since E=p²/2m). So our wave function Ψ is Ae^{i(40x - 160t)/ħ}. We can now use our operators to measure those observable quantities in our wave function.

Let's start with momentum. The operator for measuring the particle's momentum is -iħ ∂/∂x. So -iħ ∂Ψ/∂x = -iħ(i40/ħ)Ψ = 40Ψ. Since the result is a constant times Ψ, Ψ is an eigenfunction of the momentum operator with eigenvalue 40. So if we make a measurement, we will measure the momentum of the particle to be 40 kgm/s with certainty. In this case the shape of the momentum function is the same as Ψ, but its amplitude is everywhere scaled by 40.

Let's try energy. The energy operator is iħ ∂/∂t. So iħ ∂Ψ/∂t = iħ(-i160/ħ)Ψ = 160Ψ. So Ψ is an eigenfunction of the energy operator with eigenvalue 160. If we make a measurement, we will measure the energy of the particle to be 160 joules with certainty. As with momentum, the shape of the energy function is the same as Ψ, but its amplitude is scaled by 160.

Now we'll try the position. The position operator is simply x. But xΨ is not a constant times Ψ, so Ψ is not an eigenfunction of position. This means there is uncertainty about the position of the particle. The position function scales the amplitude of Ψ at each x-position by the ordinal value of that x-position (i.e., the spirals increase in amplitude along the x-axis like a cone). (Note that the eigenfunctions of the position operator are actually Dirac delta functions which spike at their respective x-positions and have zero amplitude everywhere else.)

We can actually predict these results by looking at the plane wave in Figure 1. It has a single wavelength which translates to a single momentum and energy (per the de Broglie relations). But it has the same amplitude everywhere so the particle's position is spread through spacetime. A wave function that provides a more localized position is a wave packet that combines a range of plane waves of different frequencies in superposition as shown in Figure 2.^[4] The wave packet also curls in a spiral as it propagates along the x-axis over time, but it has a localized distribution of non-zero amplitude as illustrated in Figure 2 below.^[5]

Figure 2 - Wave packet

This diagram represents a wave function at a single instant in time. Each white ball represents the complex amplitude of the wave at a particular x-position. This amplitude can be used to calculate the probability that a measurement will find the particle at that x-position.^[6] Note that x-positions beyond the two ends of the wave packet have zero amplitude which means that the particle cannot be located at those x-positions.

Suppose that the particle were prepared in a superposition of Ψ = Ψ₁ + Ψ₂ = A₁e^{i(p₁x - E₁t)/ħ} + A₂e^{i(p₂x - E₂t)/ħ}.^[7] For example, the amplitudes are A₁=6 and A₂=8 and the momenta are p₁=5 kgm/s and p₂=3 kgm/s.

If we apply the (linear) momentum operator, we get -iħ ∂/∂x(Ψ₁ + Ψ₂) = p₁Ψ₁ + p₂Ψ₂ (i.e., 5Ψ₁ + 3Ψ₂). So Ψ₁ and Ψ₂ are both eigenfunctions of the momentum operator with different momentum eigenvalues.

Now suppose a momentum of 3 kgm/s is measured in an actual experiment locating the experimenter with Ψ₂. If measurement is a linear process, the particle's momentum would also be measured to be 5 kgm/s locating the experimenter with Ψ₁.

The fact that the experimenter reports only one measurement outcome (with a probability ratio of A₁²:A₂² which is 36:64 in our example) is what gives rise to the measurement problem. The Copenhagen Interpretation postulates that the wave function collapses to Ψ₂ (Ψ = Ψ₂) and Ψ₁ disappears. Whereas the Many Worlds Interpretation assumes that the wave function Ψ continues to evolve unitarily with the experimenter and measuring apparatus now entangled with the particle in superposition.

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[1] This post builds on the concept of exponential growth in the complex number plane that was explored in Visualizing Euler's Identity.

[2] Familiar examples of wave behavior are vibrating guitar strings and ocean waves. Quantum Mechanics applies this idea to all matter via the de Broglie hypothesis and so the Schrodinger equation is a wave equation that describes how matter waves evolve. For an excellent tutorial on wave equations, see here.

Note that it is important to distinguish between the quantum state and the wave function. A wave function is a representation of a quantum state in a particular basis, such as position or momentum. This post presents the wave function in the position basis (as plane wave states of definite momentum). It can be alternatively represented in the momentum basis via a Fourier transform.

[3] Given the plane wave solution Ψ = Ae^{i(px - Et)/ħ}, the Schrodinger equation can be derived. The time derivative ∂Ψ/∂t is how the wave function changes over time and is (-iE/ħ)Ψ. The first spatial derivative ∂Ψ/∂x is how the wave function slopes in space and is (ip/ħ)Ψ. The second spatial derivative ∂²Ψ/∂x² is how the wave function curves in space and is (i²p²/ħ²)Ψ which reduces to (-p²/ħ²)Ψ.

Multiplying the time derivative by iħ gives iħ ∂Ψ/∂t = iħ(-iE/ħ)Ψ = EΨ. Multiplying the second spatial derivative by -ħ²/2m gives (-ħ²/2m)∂²Ψ/∂x² = (-ħ²/2m)(-p²/ħ²)Ψ = (p²/2m)Ψ = EΨ (E=p²/2m relates kinetic energy to momentum).

Therefore -ħ²/2m ∂²Ψ/∂x² = iħ ∂Ψ/∂t which is the time-dependent Schrodinger equation for a free particle in one dimension. The equation for any non-relativistic particle is described in Figure 3 below. Note that ∇² represents the second derivative over all space (x,y,z) and that V represents the potential energy which, for free particles, is zero.

Figure 3 - Time-dependent Schrodinger equation for a single non-relativistic particle

The Schrodinger equation expresses the principle of the conservation of energy consistent with the de Broglie relations. That is, the kinetic energy of the particle (which is proportional to the curvature of Ψ over space) plus the potential energy of the particle equals the total energy (which is proportional to the slope of Ψ over time).

The time-dependent Schrodinger equation can be more generally expressed as iħ ∂Ψ/∂t = ĤΨ where Ĥ is called the Hamiltonian operator (representing the total energy of the system) and differs with the situation or number of particles. In our free particle scenario where the potential energy is zero, Ĥ = -ħ²/2m ∂²/∂x².

The simpler time-independent Schrodinger equation applies to stationary states and is ĤΨ = EΨ where E is the total energy of the system. This is an eigenvalue equation which means that the Hamiltonian operates on the function Ψ and produces a definite (and real) energy value E multiplied by the same function Ψ. If Ψ describes the physical system and satisfies the eigenvalue equation (meaning it is an eigenfunction), then that energy eigenvalue would be measured with 100% certainty. In general, Ψ will not be an eigenfunction of the Hamiltonian but, instead, will be a linear superposition of energy eigenfunctions (with the probability of measuring a particular energy eigenvalue being the squared magnitude of the amplitude of that eigenfunction, per the Born rule).

[4] Combining a finite number of plane waves also fails to avoid the problem of the particle being delocalized since the large wave packets will still be periodic through spacetime (with other smaller periodic wave packets in between). It is only in the limit that there is a single wave packet as other wave packets would, in effect, be infinitely far away. That is, an integral over a continuous range of wave numbers (or momenta, since p=kħ) produces a single localized wave packet.

[5] Figure 2 (enlarged below) is a snapshot of a localized particle's wave function at an instant in time. Per Euler's formula, e^{i(px - Et)/ħ} = cos((px - Et)/ħ) + i.sin((px - Et)/ħ). So the complex spiral is the sum of the real cosine wave (at the back) and the imaginary sine wave (at the bottom) propagating along the x-axis. Each white ball represents the amplitude of the complex wave at that particular x-position (imagine the clock hand pointing laterally from the x-position on the x-axis to the white ball). Instead of visualizing a ball moving with the wave packet along the x-axis as time progresses, imagine that it remains at the same x-position, but simply spins around the x-axis in the complex plane, shrinking or expanding in magnitude as time progresses (i.e., as the wave packet propagates through that x-position).

Now imagine that the white ball is actually a linear combination of colored balls at that x-position, one for each plane wave in the superposition (and each with a different magnitude and phase). Each colored ball simply spins around the x-axis with a fixed magnitude but the colored balls taken together constructively and destructively interfere to produce the white ball that is seen in the image as the wave packet propagates through that x-position. That is, the entire propagating wave packet can be explained as a combination of fixed length clock hands spinning at different rates.

Figure 2 (enlarged) - Wave packet

[6] The probability that a measurement will find the particle at a particular x-position is calculated by squaring the magnitude of the wave function's amplitude at that particular x-position and time per the Born Rule. The amplitude is a complex number that, when multiplied by its complex conjugate, produces a real number that is the square of the magnitude. The squared magnitude is also the intensity of the wave function at that position and time.

[7] Ψ is a combination of different momentum basis states e^{i(p_nx - E_nt)/ħ}, each with its own coefficient (amplitude) A_n where n is the index for the basis state.