2.5 Expectation Values
Quantum mechanics is probabilistic by its very nature. So in general the results of individual measurements cannot be precisely predicted. And we can make very specific predictions about average measurement outcomes. Those predictions come in the form of the "expectation value" of an observable. We use following equation to determine the expectation value of an observable (𝑜) represented by the operator O for a system in state in ∣𝜓⟩:
(2.55) ⟨𝑜⟩ = ⟨𝜓∣ O ∣𝜓⟩.
The mesurements are made on group of system (usually called an "ensemble" of systems) all prepared to be in the same state prior to the measurement.
Consider a soccer matching analogy and for example we have following the table of the probabilities of scoring one to six goals:
[Winning side total goals - Probability (%)] ⇔ (0 - 0) (1 - 22) (2 - 43) (3 - 18) (4 - 9) (5 - 5) (6 - 3)
Given this information, the expected number of goals (⟨𝑔⟩) for the winning team can be determined simply by multiplying each possible score (call it 𝜆𝑛) by its probability (𝑃𝑛) and summing the result over all possible score:
(2.56) ⟨𝑔⟩ = ∑𝑛=1𝑁𝜆𝑛𝑃𝑛,
so in this case we obtain
⟨𝑔⟩ = 𝜆0𝑃0 + 𝜆1𝑃1 + 𝜆2𝑃2 + 𝜆3𝑃3 + 𝜆4𝑃4 + 𝜆5𝑃5 + 𝜆6𝑃6 = 0(0) + 1(0.22) + 3(0.18) + 4(0.09) + 5(0.05) + 6(0.03) = 2.4
Similarly, in quantum mechanics, consider a Hermitian operator O and normalized wavefunction represented by the ket ∣𝜓⟩. As explained in Section 1.6, the ket can be written as the weighted combination of kets representing the eigenvectors of operator O:
(1.35) ∣𝜓⟩ = 𝑐1 ∣𝜓1⟩ + 𝑐2 ∣𝜓2⟩ + ⋅ ⋅ ⋅ + 𝑐𝑁 ∣𝜓𝑁⟩ = ∑𝑛=1𝑁𝑐𝑛 ∣𝜓𝑛⟩.
where 𝑐1 through 𝑐𝑁 represent the amount of each orthonormal eigenfunction ∣𝜓𝑛⟩. Now consider ⟨𝜓∣ O ∣𝜓⟩ and we get
(2.57) O ∣𝜓⟩ = O ∑𝑛=1𝑁𝑐𝑛 ∣𝜓𝑛⟩ = ∑𝑛=1𝑁𝑐𝑛 O ∣𝜓𝑛⟩ = ∑𝑛=1𝑁𝜆𝑛𝑐𝑛 ∣𝜓𝑛⟩, 𝜆𝑛: eigenvalue of O applied to ∣𝜓𝑛⟩.
⟨𝜓∣ = ⟨𝜓1∣𝑐*1 + ⟨𝜓2∣𝑐*2 + ⋅ ⋅ ⋅ + ⟨𝜓𝑁∣𝑐*𝑁 = ∑𝑚=1𝑁⟨𝜓𝑚∣𝑐*𝑚, ⟨𝜓∣ O ∣𝜓⟩ = ∑𝑚=1𝑁⟨𝜓𝑚∣𝑐*𝑚∑𝑛=1𝑁𝜆𝑛𝑐𝑛 ∣𝜓𝑛⟩ = ∑𝑚=1𝑁∑𝑛=1𝑁𝑐*𝑚𝜆𝑛𝑐𝑛 ⟨𝜓𝑚∣𝜓𝑛⟩
But if the eigenfunction 𝜓𝑛 are orthonormal, only the terms with 𝑛 = 𝑚 survives, so this becomes
(2.58) ⟨𝜓∣ O ∣𝜓⟩ = ∑𝑛=1𝑁𝜆𝑛𝑐*𝑛𝑐𝑛 = ∑𝑛=1𝑁𝜆
𝑛∣𝑐𝑛∣2 = ⟨𝑜⟩, ∣𝑐𝑛∣2: probability of obtaining result 𝜆𝑛.
This expression can be extended to the case by a continuous variable 𝑥 rather than discrete values 𝜆𝑛. The expectation value of the observable 𝑥 is then
(2.59) ⟨𝑥⟩ = ∫-∞∞𝑥𝑃(𝑥)𝑑𝑥.
Using the inner product, the expectation value can be written in Dirac notation and integral form as
(2.60) ⟨𝑥⟩ = ⟨𝜓∣ 𝑋^ ∣𝜓⟩ = ∫-∞∞[𝜓(𝑥)]*𝑋^[𝜓(𝑥)] 𝑑𝑥.∣𝑑𝑥, 𝑋^: the oprator associated with observable 𝑥.
In quantum mechanics, calling the uncertainty in position 𝛥𝑥, the square of the uncertainty is given by ※
(2.61, 62) (𝛥𝑥)2 = ⟨𝑥2⟩ - ⟨𝑥⟩2, 𝛥𝑥 = √[⟨𝑥2⟩ - ⟨𝑥⟩2],
where ⟨𝑥⟩ represents the expectation value of 𝑥 and ⟨𝑥2⟩ represents the expectation value of 𝑥2.
Also (𝛥𝑥)2 is equivalent to the variance of 𝑥 and the variance is defined as following
(2.63) Variance of 𝑥 = (𝛥𝑥)2 ≡ ⟨(𝑥 - ⟨𝑥⟩)2⟩,
which means 𝛥𝑥, square root of the variance, is the standard deviation of the distribution 𝑥. Thus the uncertainty in position is given by Eq. 2.62. Similarly, the uncertainty in momentum 𝛥𝑝 and 𝛥𝐸 respectively given by ※
(2.63, 64) 𝛥𝑝 = √[⟨𝑝2⟩ - ⟨𝑝⟩2], 𝛥𝐸 = √[⟨𝐸2⟩ - ⟨𝐸⟩2],
3. The Schrödinger Equation .
The Schrödinger Equation, which was published in 1926, describes how a quantum state evolve over time, and understanding the physical meaning of the terms of the equation prepare we to understand the behavior of quantum wavefunctions.
3.1 Origin of the Schrödinger Equation
There are several ways to "derive" the Schrödinger equation, and none of those methods are rigorous derivation from first principles!
From the start Erwin Schrödinger(1887-1961) clearly recognized the need for the quantum wave equation which should be the form of first-order in time and different from the second-order partial differential equation in both space and time by French physicist Louis de Broglie. He also saw the making the equation complex provided immense benefits.
We begin with Plank-Einstein relation of the energy of a photon (𝐸) to its frequency (𝑓) or angular frequency (𝜔 = 2π𝑓):
(3.1) 𝐸 = 𝘩𝑓 = ℏ𝜔.
where 𝘩 represents the Planck constant and ℏ is the modified Plank constant (ℏ = 𝘩/2π).
Another useful equation comes from Maxwell's radiation pressure in 1862. The magnitude of the momentum of electromagnetic waves (𝑝) is related to energy (𝐸) and the speed of light (𝑐) by the relation
(3.2) 𝑝 = 𝐸/𝑐.
In 1924, de Broglie suggested that at the quantum level particles can exhibit wavelike behavior and the momentum of these "matter waves" can be determined by the above equations
(3.3) 𝑝 = 𝐸/𝑐 = ℏ𝜔/𝑐.
Sinthe the frequency (𝑓) of a wave is related to its wavelength (𝜆) amd speed (𝑐) by the equation 𝑓 = 𝑐/𝜆, the momentum is
𝑝 = ℏ𝜔/𝑐 = ℏ(2π𝑓)/𝑐 = ℏ(2π𝑐/𝜆)/𝑐 = ℏ2π/𝜆.
The definition of wavenumber (𝑘 ≡ 2π/𝜆) makes this
(3.4) 𝑝 = ℏ𝑘.
This equation is the de Broglie's relation and it represents the mixing of wave and particle behavior into the concept of "wave-particle duality".
Since 𝑝 = 𝑚𝑣 in the nonrelativistic case, the classical equation for kinetic energy is
(3.5) 𝐾𝐸 = (1/2)𝑚𝑣2 = 𝑝2/𝑚,
(3.6) 𝐾𝐸 = ℏ2𝑘2/2𝑚.
Now from Eq. 3.1 the total energy (𝐸) as the sum of the kinetic energy (𝐾𝐸) and potential energy (𝑉) :
(3.8) 𝐸 = ℏ𝜔 = ℏ2𝑘2/2𝑚 + 𝑉.
This equation provides the foundation for the Schrödinger equation for a quantum wavefunction 𝛹(𝑥, 𝑡).
To get from Eq. 3.8 to the Schrodinger equation, one path is to assume that the quantum wavefunction has the form of a plane wave, for which the surfaces of constant phase are flat plane. For a plane wave propagating in the positive 𝑥direction, the wavefunction is given by ※
(3.9) 𝛹(𝑥, 𝑡) = 𝛢𝑒𝑖(𝑘𝑥-𝜔𝑡),
in which 𝛢 represents the wave's amplitude, 𝑘 is the wavenumber, and 𝜔 is the angular frequency of the wave.
With this expression for 𝛹, taking temporal and spatial derivatives is straightforward as follows
(3.10) ∂𝛹(𝑥, 𝑡)/∂𝑡 = ∂[𝛢𝑒𝑖(𝑘𝑥-𝜔𝑡)]/∂𝑡 = -𝑖𝜔[𝛢𝑒𝑖(𝑘𝑥-𝜔𝑡)] = -𝑖𝜔𝛹(𝑥, 𝑡).
So we have
(3.11) ∂𝛹/∂𝑡 = -𝑖𝜔𝛹,
which means we can write 𝜔 as
(3.12) 𝜔 = 1/(-𝑖𝛹) ∂𝛹/∂𝑡 = 𝑖 (1/𝛹) ∂𝛹/∂𝑡. [∵ 1/𝑖 = -(𝑖)(𝑖)/𝑖 = -𝑖]
If we take the first partial derivative of 𝛹(𝑥, 𝑡) with respective to space (𝑥):
(3.13) ∂𝛹(𝑥, 𝑡)/∂𝑥 = ∂[𝛢𝑒𝑖(𝑘𝑥-𝜔𝑡)]/∂𝑥 = 𝑖𝑘[𝛢𝑒𝑖(𝑘𝑥-𝜔𝑡)] = 𝑖𝑘𝛹(𝑥, 𝑡)
So we have
(3.14) ∂𝛹/∂𝑥 = 𝑖𝑘𝛹,
It is also helpful to take the second partial derivative of the function, which gives
(3.15) ∂2𝛹(𝑥, 𝑡)/∂𝑥2 = ∂[𝑖𝑘𝛢𝑒𝑖(𝑘𝑥-𝜔𝑡)]/∂𝑥 = -𝑘2𝛹(𝑥, 𝑡),
So we have
(3.16, 17) ∂2𝛹/∂𝑥2 = -𝑘2𝛹, 𝑘2 = -(1/𝛹) ∂2𝛹/∂𝑥2
Now the total energy equation (3.8) from Eq. 3.12 and 3.17 becomes
(3.18) 𝐸 = ℏ𝜔 = ℏ[𝑖 (1/𝛹) ∂𝛹/∂𝑡] = 𝑖ℏ(1/𝛹) ∂𝛹/∂𝑡, [left hand side]
(3.19) 𝐸 = ℏ2𝑘2/2𝑚 + 𝑉 = ℏ2/2𝑚[-(1/𝛹) ∂2𝛹/∂𝑥2] + 𝑉, [right hand side] and so we have
(3.20) 𝑖ℏ (1/𝛹) ∂[𝛹(𝑥, 𝑡)]/∂𝑡 = -(ℏ2/2𝑚) (1/𝛹) ∂2[𝛹(𝑥, 𝑡)]/∂𝑥2 + 𝑉, and multiplying by wavefunction yields
(3.21) 𝑖ℏ ∂𝛹/∂𝑡 = -(ℏ2/2𝑚) ∂2𝛹/∂𝑥2 + 𝑉𝛹 or 𝑖ℏ ∂[𝛹(𝑥, 𝑡)]/∂𝑡 = -(ℏ2/2𝑚) ∂2[𝛹(𝑥, 𝑡)]/∂𝑥2 + 𝑉[𝛹(𝑥, 𝑡)]
This is the most common form of one-dimensional time-dependent Schrodinger equation. We have used the total energy relation and the expression for a plane wave. Why should we expect this equation to hold for quantum wavefunctions of other forms.
One answer is this: it works! It seems surprising that this equation leads to prediction that agree with laboratory measurements of quantum observables such as position, momentum and energy. Moreover since the Schrödinger equation is linear, the superposition of plane waves are also solutions conveniently.
From a finite number (𝑁) of discrete plane wave components, a wave linear combination may be used:
(3.22) 𝛹(𝑥, 𝑡) = 𝛢1𝑒𝑖(𝑘1𝑥-𝜔1𝑡) + 𝛢2𝑒𝑖(𝑘2𝑥-𝜔2𝑡) + ⋅ ⋅ ⋅ + 𝛢𝑁𝑒𝑖(𝑘𝑁𝑥-𝜔𝑁𝑡) = ∑𝑛=1𝑁𝛢𝑛𝑒𝑖(𝑘𝑛𝑥-𝜔𝑛𝑡).
Alternatively the wavefunction can be synthesized using a continuous spectrum of plane waves:
(3.23) 𝛹(𝑥, 𝑡) = ∫-∞∞ 𝛢(𝑘)𝑒𝑖(𝑘𝑥-𝜔𝑡) 𝑑𝑘,
where 𝛢(𝑘) represents the amplitude per unit wavenumber. And a very common and useful version can be obtained by a constant factor of 1/√(2π) out of the wavefunction
𝛢(𝑘) and setting the time to an initial reference time (𝑡 = 0):
(3.24) 𝛹(𝑥, 0) = 1/√(2π) ∫-∞∞ 𝜙(𝑘)𝑒𝑖𝑘𝑥 𝑑𝑘.
Another version of Eq. 21 looks like this:
(3.25) 𝑖ℏ ∂𝛹/∂𝑡 = Ĥ𝛹,
where Ĥ represents the "Hamiltonian" or the total-energy operator. So it gives
Ĥ𝛹 = -(ℏ2/2𝑚) ∂2𝛹/∂𝑥2 + 𝑉𝛹,
which means that the Hamiltonian operator is equivalent to
(3.26) Ĥ = -(ℏ2/2𝑚) ∂2/∂𝑥2 + 𝑉.
Since 𝐸 = ℏ𝜔 and 𝑝 = ℏ𝑘, we can rewrite Eq. 3.9
(3.27) 𝛹(𝑥, 𝑡) = 𝛢𝑒𝑖(𝑘𝑥-𝜔𝑡) = 𝛢𝑒𝑖[(𝑝/ℏ)𝑥-(𝐸/ℏ)𝑡] = 𝛢𝑒𝑖/ℏ(𝑝𝑥-𝐸𝑡),
and then take the first-order spatial derivative:
(3.28) ∂𝛹/∂𝑥 = [(𝑖/ℏ)𝑝]𝛢𝑒𝑖/ℏ(𝑝𝑥-𝐸𝑡) = (𝑖/ℏ)𝑝𝛹 or 𝑝𝛹 = (𝑖/ℏ) ∂𝛹/∂𝑥 = -𝑖ℏ ∂ 𝛹/∂𝑥.
This suggests the (one-dimensional) differential operator associated with momentum may be written as
(3.29) 𝑃^ = -𝑖ℏ ∂/∂𝑥.
Since the square of an operator is formed by applying the operator twice,
(3.31) (𝑃^)2𝛹 = 𝑃^(𝑃^𝛹) = -𝑖ℏ ∂/∂𝑥(-𝑖ℏ ∂/∂𝑥) = 𝑖2ℏ2 ∂2𝛹/∂𝑥2 = -ℏ2 ∂2𝛹/∂𝑥2.
Thus the (𝑃^)2 operator may be written as
(3.32) (𝑃^)2 = -ℏ2 ∂2/∂𝑥2
and plugging this expression into Eq. 3.30 gives
(3.30) Ĥ = (𝑃^)2/2𝑚 + 𝑉.
In addition, there are some alternative approaches to "derive" the Schrödinger equation, for example, "probability flow" approach and "path integral" approach.
3.2 What the Schrödinger Equation Means
It's worthwhile to step back to ask "What is the Schrödinger equation telling me?". So we will review a brief description of each term and the dimensions and its SI units of the following equation
𝑖ℏ ∂𝛹/∂𝑡 = -(ℏ2/2𝑚) ∂2𝛹/∂𝑥2 + 𝑉𝛹
• ∂𝛹/∂𝑡 : The quantum wavefunction 𝛹(𝑥, 𝑡) is a function of both time and space and in a graph of the wavefunction at a given location, this term is the slope of the graph in time. The one-dimensional quantum wavefunction 𝛹 represents a probability density amplitude※ and the square of which has dimensions of probability per unit length. Since provability is dimensionless this is equivalent to 1/m in the SI system. The 𝛹 must have units of 1/√m and ∂𝛹/∂𝑡 has SI units of 1/(s√m).
• 𝑖 : As an operator, multiplication by 𝑖 has the effect of causing a 90∘ rotation in the complex plane (Fig. 1.7), moving numbers from the positive real axis to the positive imaginary axis or from the positive imaginary axis to the negative real axis, for example. The factor 𝑖 is, of course, dimensionless.
• ℏ : The modified Planck constant ℏ is the Planck constant 𝘩 devided by 2π. Just as 𝘩 is the constant of proprtionality between the energy (𝐸) and frequency (𝑓) of a photon (𝐸 = 𝘩𝑓), ℏ is the correspondence between total energy (𝐸) and angular frequency (𝜔) and between momentum (𝑝) and wavenumber (𝑘) in quantum wavefunctions, as in the equation 𝐸 = ℏ𝜔 and 𝑝 = ℏ𝑘 which yield 𝐾𝐸 = (ℏ𝑘)2/2𝑚.
Since the SI unit of energy is the Joule (J = m2kg /s2) and that of frequency is Hertz, which is defined as one cycle per second, is 1/s, the SI units of 𝘩 are Joule per Hertz (Js or m2kg/s). So ℏ has the dimensions of Joules per Hertz per radian (Js/rad or m2kg /(s2res)). The numerical values of 𝘩 and ℏ in the SI system are 𝘩 = 6.62607 ⨯ 10-34 Js and ℏ = 1.05457 ⨯ 10-34 Js/rad.
• 𝑚 : the mass of the particle or system associated with 𝛹(𝑥, 𝑡) is a inertia, that is, resistance to acceleration. In the SI system mass has units of kilogrma (kg).
• ∂2𝛹/∂𝑥2 : ∂𝛹/∂𝑥 gives the change of the wavefunction over space (the slope of the wavefunction plotted against 𝑥), and ∂2𝛹/∂𝑥2 gives the change in the slope (the curvature) of the wavefunction over space. Since 𝛹(𝑥, 𝑡) has SI units of 1/√m, ∂2𝛹/∂𝑥2 has units of 1/(m2√m) = 1/m5/2.
• 𝑉 : the potential energy of the system may vary over space and time, where we will see 𝑉(𝑥, 𝑡) for one dimensional case or 𝑉(𝑟̄, 𝑡) in the three dimensional case. Unlike classical mechanics, where potential, kinetic and total energy have precise values, in quantum mechanics only the average or expectation value of energy may be determined., and a particle's total energy may be less than the potential energy in some regions. The potential-energy term has dimension of energy and SI units of Joules (equivalent to kg m2/s2).
So the terms of the Schrödinger equation form a parbolic second-order partial differential equation. Because the combination of ∂𝛹/∂𝑡 and ∂2𝛹/∂𝑥2 is analogous to the parabolic equation 𝑦 = 𝑐𝑥2, that is
(3.33) ∂[𝑓(𝑥, 𝑡)]/∂𝑡 = 𝐷 ∂2[𝑓(𝑥, 𝑡)]/∂𝑥2.
This is the one-dimensional "diffusion" equation which is also called heat equation and describes the behavior of a quantity 𝑓(𝑥, 𝑡) with spatial distribution that may evolve over time, such as the concentration of a substance or the temperature of a fluid and "D" represents the diffusion coefficient.
Consider the case in which the potential energy (𝑉) is zero, then we have
(3.34) ∂[𝛹(𝑥, 𝑡)]/∂𝑡 = 𝑖ℏ2/2𝑚 ∂2[𝛹(𝑥, 𝑡)]/∂𝑥2.
We can see the function 𝑓(𝑥, 𝑡) shown in Fig. 3.3 where this function has a maximum at 𝑥 = 0 and inflection points are 𝑥 = -3 and 𝑥 = +3 for waveform at time 𝑡 = 0 in real line.
For the region to the left of the inflection point at 𝑥 = -3, the slope of the function (∂𝑓/∂𝑥) is positive and getting more positive as 𝑥 increases, which means the curvature (the change of slope, ∂2𝑓/∂𝑥2) is positive. Likewise, to the right of 𝑥 = +3, the slope is negative and but getting less negative with increasing 𝑥, meaning that the curvature is positive as well. But between 𝑥 = -3 and 𝑥 = 0, the slope is positive and the curvature is negative (getting less steep). And between 𝑥 = 0 and 𝑥 = +3, the slope is negative and the curvature is negative (getting steeper).
And the diffusion equation tells us that the time rate of change of 𝑓(𝑥, 𝑡) is proportional to the curvature of the function, the function will evolve as shown in Fig. 3.3 in dot line. If 𝑓(𝑥, 𝑡) represents temperature, for example, we would expect the energy from the initially warm region diffuses into cooler neighboring regions.
Though given the similarity between the Schrödinger equation and the classical diffusion equation, because of the factor "𝑖" in the Schrödinger equation, which means that 𝛹 can be complex, the quantum particles and systems diffuse sometimes, but not always as time passes and may exhibit wavelike (oscillatory) behavior under some circumstances.
According to the "Born rule" (by Max Born in 1926) which is now widely accepted, 𝛹 can be interpreted as a probability amplitude and the modulus squared (∣𝛹∣2 = 𝛹*𝛹) of a particle's position-space wavefunction gives the particle's position probability density (the provability per unit length in the one-dimensional case). This means that the integral of position probability density function over any spatial interval gives the probability of finding the particle within that interval.
So although the Schrödinger equation can't be derived from first principles, it gives results that predict and describe the behavior of quantum particles and systems over space and time.
3.3 Time-Independent Schrödinger Equation
Separating out the time-dependent and space-dependent terms of the Schrödinger equation is helpful in understanding why the quantum wavefunction behaves as it does.
This technique often works and in any situation in which the potential energy varies only over space (and not over time), we can use separation of variables to solve the Schrödinger equation.
So start by writing the quantum wavefunction as the product of 𝛹(𝑥) (depends only on space) and 𝑇(𝑡):
(3.35) 𝛹(𝑥, 𝑡) = 𝜓(𝑥)𝑇(𝑡).
Inserting this into the Schrödinger equation gives
𝑖ℏ ∂[𝜓(𝑥)𝑇(𝑡)]/∂𝑡 = -(ℏ2/2𝑚) ∂2[𝜓(𝑥)𝑇(𝑡)]/∂𝑥2 + 𝑉[𝜓(𝑥)𝑇(𝑡)].
𝑖ℏ 𝜓(𝑥)𝑑[𝑇(𝑡)]/𝑑𝑡 = -(ℏ2/2𝑚) 𝑇(𝑡)𝑑2[𝜓(𝑥)]/𝑑𝑥2 + 𝑉[𝜓(𝑥)𝑇(𝑡)].
Then divide the both sides of the equation by 𝜓(𝑥)𝑇(𝑡) and we obtain:
[1/𝜓(𝑥)𝑇(𝑡)]𝑖ℏ 𝜓(𝑥)𝑑[𝑇(𝑡)]/𝑑𝑡 = -[1/𝜓(𝑥)𝑇(𝑡)] (ℏ2/2𝑚) 𝑇(𝑡)𝑑2[𝜓(𝑥)]/𝑑𝑥2 + [1/𝜓(𝑥)𝑇(𝑡)]𝑉[𝜓(𝑥)𝑇(𝑡)],
(3.36) 𝑖ℏ[1/𝑇(𝑡)]𝑑[𝑇(𝑡)]/𝑑𝑡 = -(ℏ2/2𝑚) [1/𝜓(𝑥)] 𝑑2[𝜓(𝑥)]/𝑑𝑥2 + 𝑉.
Now its left side a function only of time and the right side is a function only of location. For that to be true, each side must be constant. That is, while the left side is changing over time, the right side would not be changing and vice versa. So for this equation to hold true, both sides must be constant.
So what does it mean? Look first at the left side:
𝑖ℏ[1/𝑇(𝑡)]𝑑[𝑇(𝑡)]/𝑑𝑡 = (constant)
(3.37) [1/𝑇(𝑡)] 𝑑[𝑇(𝑡)]/𝑑𝑡 = (constant)/𝑖ℏ.
Integrating the both side gives
∫0𝑡[1/𝑇(𝑡)] 𝑑[𝑇(𝑡)]/𝑑𝑡 = ∫0𝑡 (constant)/𝑖ℏ 𝑑𝑡 ln[𝑇(𝑡)] = [(constant)/𝑖ℏ]𝑡 = -𝑖[(constant)/ℏ]𝑡 𝑇(𝑡) = 𝑒-𝑖[(constant)/ℏ]𝑡
Calling the constant 𝐸 makes this
(3.38) 𝑇(𝑡) = 𝑒-𝑖(𝐸/ℏ)𝑡.
So since 𝑑[𝑇(𝑡)]/𝑑𝑡 = -𝑖(𝐸/ℏ)𝑇(𝑡), 𝑖ℏ[1/𝑇(𝑡)]𝑑[𝑇(𝑡)]/𝑑𝑡 = 𝐸, we have have
(3.39) -(ℏ2/2𝑚) [1/𝜓(𝑥)] 𝑑2[𝜓(𝑥)]/𝑑𝑥2 + 𝑉 = 𝐸.
Multiplying all terms by 𝛹(𝑥) gives
(3.40) -(ℏ2/2𝑚) 𝑑2[𝜓(𝑥)]/𝑑𝑥2 + 𝑉[𝜓(𝑥)] = 𝐸[𝜓(𝑥)].
This equation is called the time-independent Schrödinger equation (TISE), since its solutions 𝜓(𝑥) describe only the spatial behavior of quantum wavefunction 𝛹(𝑥, 𝑡) (the temporal behavior is described by the 𝑇(𝑡) functions).
This is an eigenvalue equation. To see that, consider the operator
(3.41) Ĥ = -(ℏ2/2𝑚) ∂2/∂𝑥2 + 𝑉.
As mentioned in Section 3.1, this is the one-dimensional version of the Hamiltonian (total energy) operator. Using this in the TISE makes it look like this:
(3.42) Ĥ[𝜓(𝑥)] = 𝐸[𝜓(𝑥)],
which is an eigenvalue equation, with eigenfunction 𝜓(𝑥) and eigenvalue 𝐸. This is why many authors refers to the process of solving the TISE as "finding the eigenvalue and eigenfunctions of the Hamiltonian operator."
If any wavefunction 𝛹(𝑥, 𝑡) that may be separated into spatial and temporal function (as 𝛹(𝑥, 𝑡) = 𝜓(𝑥)𝑇(𝑡) in Eq. 3.35), quantities such as the probability density and expectation values do not vary over time, so the functions 𝜓(𝑥) is called in "stationary states". To see why that's true, observe what happen when we form the inner product of such a separable wavefunction with itself:
⟨𝛹(𝑥, 𝑡)∣𝛹(𝑥, 𝑡)⟩ ∝ 𝛹*𝛹 = [𝜓(𝑥)𝑇(𝑡)]*[𝜓(𝑥)𝑇(𝑡)] = [𝜓(𝑥)𝑒-𝑖(𝐸/ℏ)𝑡]*[𝜓(𝑥)𝑒-𝑖(𝐸/ℏ)𝑡] = [𝜓(𝑥)]*𝑒𝑖(𝐸/ℏ)𝑡[𝜓(𝑥)]𝑒-𝑖(𝐸/ℏ)𝑡 = [𝜓(𝑥)]*[𝜓(𝑥)],
in which the time dependence has disappeared. Hence any quantity involving 𝜓*𝜓 will not change over time (it will become "stationary") whenever 𝛹(𝑥, 𝑡) is separable.
3.4 Three-Dimensional Schrödinger Equation
As we have surmised, extending the Schrödinger equation to three dimensions involves writing the wavefunction as 𝛹(𝑟̄, 𝑡), rather than 𝛹(𝑥, 𝑡).
In 3-D Cartesian coordinates, the position vector ř can be expressed using the orthonormal basic vectors (𝑖̄, 𝑗̄, 𝑘̄):
(3.43) 𝑟̄ = 𝑥𝑖̄ + 𝑦𝑗̄ + 𝑧𝑘̄.
Likewise, in the 3-D case, the wavenumber becomes a vector 𝑘̄, which can be expressed with vector components 𝑘x, 𝑘y, and 𝑘z as
(3.44) ǩ = 𝑘x𝑖̄ + 𝑘y𝑗̄ + 𝑘z𝑘̄
(3.45) ∣ǩ∣ = √(𝑘x2 + 𝑘y2 + 𝑘z2) = 2π/𝜆.
(3.46) 𝛹(𝑟̄, 𝑡) = 𝐴𝑒𝑖(ǩ⋅𝑟̄-𝜔𝑡),
where ǩ⋅𝑟̄ represents the scalar product between vectors 𝑟̄ and ǩ. We can take a look at Fig. 3.5.
We may see that do product expanded in Cartesian coordinates as
ǩ ⋅ 𝑟̄ = (𝑘x𝑖̄ + 𝑘y𝑗̄ + 𝑘z𝑘̄) ⋅ (𝑥𝑖̄ + 𝑦𝑗̄ + 𝑧𝑘̄) = 𝑘x𝑥 + 𝑘y𝑦 + 𝑘z𝑧, so we have
(3.47) 𝛹(𝑟̄, 𝑡) = 𝐴𝑒𝑖(𝑘x𝑥 + 𝑘y𝑦 + 𝑘z𝑧)-𝜔𝑡).
It's also necessary to extend ∂2/∂𝑥2 to three dimensions.
∂𝛹(𝑟̄, 𝑡)/∂𝑥 = ∂[𝐴𝑒𝑖(𝑘x𝑥 + 𝑘y𝑦 + 𝑘z𝑧)-𝜔𝑡)]/∂𝑥 = 𝑖𝑘x𝛹(𝑟̄, 𝑡), ∂2𝛹(𝑟̄, 𝑡)/∂𝑥2 = ∂[𝑖𝑘x𝐴𝑒𝑖(𝑘x𝑥 + 𝑘y𝑦 + 𝑘z𝑧)-𝜔𝑡)]/∂𝑥 = -𝑘x2𝛹(𝑟̄, 𝑡)
We can get the second spatial derivatives with respect to 𝑦 and 𝑧 and adding them together gives
∂2𝛹(𝑟̄, 𝑡)/∂𝑥2 + ∂2𝛹(𝑟̄, 𝑡)/∂𝑦2 + ∂2𝛹(𝑟̄, 𝑡)/∂𝑧2 = -(𝑘x2 + 𝑘y2 + 𝑘z2)𝛹(𝑟̄, 𝑡).
From Eq. 3.45, we obtain
(3.48) ∂2𝛹(𝑟̄, 𝑡)/∂𝑥2 + ∂2𝛹(𝑟̄, 𝑡)/∂𝑦2 + ∂2𝛹(𝑟̄, 𝑡)/∂𝑧2 = -∣ǩ∣2𝛹(𝑟̄, 𝑡).
Comparing this equation to Eq. 3.16 shows that the sum of the second spacial derivatives bring down to -∣ǩ∣2 from the plane-wave exponential just as -𝑘2 in the one dimensional case.
This sum of the second spatial derivatives can be written as a differential operator, Laplacian operator
∂2𝛹(𝑟̄, 𝑡)/∂𝑥2 + ∂2𝛹(𝑟̄, 𝑡)/∂𝑦2 + ∂2𝛹(𝑟̄, 𝑡)/∂𝑧2 = (∂2/∂𝑥2 + ∂2/∂𝑦2 + ∂2/∂𝑧2)𝛹(𝑟̄, 𝑡)
(3.49) 𝛻2 = ∂2/∂𝑥2 + ∂2/∂𝑦2 + ∂2/∂𝑧2
With the Laplacian operator 𝛻2 and the 3-dimensional wavefunction 𝛹(𝑟̄, 𝑡) in hand, we can write the Schrödinger equation as
(3.50) 𝑖ℏ ∂𝛹(𝑟̄, 𝑡)/∂𝑡 = -(ℏ2/2𝑚) 𝛻2𝛹(𝑟̄, 𝑡) + 𝑉[𝛹(𝑟̄, 𝑡)].
Referring to Eq. 3.33, the 3-dimensional version of the diffusion equation is
(3.51) ∂[𝑓(𝑟̄, 𝑡)]/∂𝑡 = 𝐷 𝛻2[𝑓(𝑟̄, 𝑡)]𝛻2.
And in the case of which the potential energy (𝑉) is zero and we can write Eq. 3.50 as
(3.52) ℏ ∂𝛹(𝑟̄, 𝑡)/∂𝑡 = (𝑖ℏ2/2𝑚) 𝛻2𝛹(𝑟̄, 𝑡).
As in 1-D case, the presence of "𝑖" factor in the Schrödinger equation has important implications, but the fundamental relationship in both of equations os this: the evolution of the wavefunction over time is proportional to the Laplacian of the wavefunction.
To understand the nature of the Laplacian, it helps to view spatial curvature from the perspective that is to consider how the value of a function at a given point compares to the average value of that function at equidistant neighboring points as we can see in Fig. 3.6. In other words, the curvature of a function at any location is a amount by which the vale of the function at that location equals, exceeds, or falls short of the average value of the function at surrounding points.
We may remind that the mathematical definition of gradient operator in 3-D Cartesian coordinates using "del" is:
(3.53) 𝛻̄ = 𝑖̄(∂/∂𝑥) + 𝑗̄(∂/∂𝑦) + 𝑘̄(∂/∂𝑧).
So the gradient of 𝜓(𝑥, 𝑦, 𝑧) in Cartesian coordinates is
(3.54) 𝛻̄𝜓(𝑥, 𝑦, 𝑧) = (∂𝜓/∂𝑥)𝑖̄ + (∂𝜓/∂𝑦)𝑗̄ + (∂𝜓/∂𝑧)𝑘̄.
And the divergence operator which works in tandem with the gradient to produce the Laplacian is written a a scalar product between the gradient operator and a vector (such as 𝐴̄). In 3-D Cartesian coordinates, that means
(3.55) 𝛻̄ ⋅ 𝐴̄ = [𝑖̄(∂/∂𝑥) + 𝑗̄(∂/∂𝑦) + 𝑘̄(∂/∂𝑧)] ⋅ (𝐴x𝑖̄ + 𝐴y𝑗̄ + 𝐴z𝑘̄) = ∂𝐴x/∂𝑥 + ∂𝐴y/∂𝑦 + ∂𝐴z/∂𝑧.
It's worth noting that a 3-D version of the time-independent Schrödinger equation (TISE) can be found using an approach similar to that used in 1-D case in Eq. 3.35. So we have
(3.56) 𝛹(𝑟̄, 𝑡) = 𝜓(𝑟̄)𝑇(𝑡),
and write a 3-D version of the potential energy 𝑉(𝑟̄). Just as in the 1-D casw, the time portion is 𝑇(𝑡) = 𝑒-𝑖(𝐸/ℏ)𝑡, but the 3-D spatial portion equation is
(3.57) -(ℏ2/2𝑚) 𝛻̄2[𝜓(𝑟̄)] + 𝑉[𝜓(𝑟̄)] = 𝐸[𝜓(𝑟̄)].
In this case the 3-D version of the Hamiltonian operator is
(3.58) Ĥ = -(ℏ2/2𝑚) 𝛻̄2 + 𝑉.
One final note on Laplacian operator that appears in the 3-D Schrödinger equation: with spherical symmetry, it may be easier to apply the Laplacian operator in spherical coordinates. That version looks like this:
(3.59) 𝛻̄2 = (1/𝑟2) (∂/∂𝑟)[𝑟2(∂/∂𝑟)] + [1/(𝑟2 sin 𝜃)](∂/∂𝜃)[sin 𝜃 (∂/∂𝜃)] + [1/(𝑟2 sin2 𝜃)](∂2/∂𝜙2).
※ attention: some rigorous derivation might be required
