What does the second derivative actually do in math and physics?

Summary

视频脚本深入探讨了物理学家费曼关于“球面上的平均值”的概念，通过直观地理解二阶导数在数学和物理学中的作用，揭示了它在薛定谔方程、电磁学等领域的重要性。视频首先回顾了一阶导数的直观意义，即函数随变量微小变化的速率。随后，通过费曼的“平均在球上”的概念，引入了二阶导数，并解释了它与函数曲率的关系。通过一维和三维的类比，视频推导出了二阶导数的几何意义，并将其应用于电势能函数的推导，直观地导出了麦克斯韦方程之一。此外，视频还联系了量子力学中的动能算符，展示了如何通过二阶导数理解量子现象，尤其是海森堡不确定性原理。最后，视频鼓励观众利用这一理解去构建对其他物理现象的新直觉。

Takeaways

• 📚 费曼在1964年康奈尔大学的讲座中提出了一个概念，即通过了解一个小球体表面的物理量，而不需要考虑外部情况，来理解该点的势能。
• 🎓 视频旨在深入探讨费曼的“球面上的平均值”概念，并发展对数学和物理中二阶导数的直观理解。
• 🔗 哈佛量子计划的量子短片大赛为不同背景的人提供了一个展示量子物理知识的机会，参赛者可以通过视频解释量子物理的某个方面。
• 📈 一阶导数直观地告诉我们，当x发生微小变化时，函数f(x)的变化量。
• 📉 二阶导数通常被理解为一阶导数随输入变化的变化率，但视频中提供了一种更直观的理解方式，即函数在一点的曲率。
• 🧮 通过泰勒级数展开，可以量化函数在x0点附近值的平均值与x0点值的差，这与二阶导数有关。
• 🟢 函数的曲率可以通过“球面上的平均值”来量化，这有助于我们理解二阶导数。
• 🟠 在三维空间中，类似的概念涉及在一个小球面上计算函数的平均值，并与三维空间的二阶导数（拉普拉斯算子）相关。
• ⚡ 利用对二阶导数的直观理解，可以推导出电荷分布产生的电势能函数的微分方程，这与麦克斯韦方程组之一相符。
• 🌌 在量子力学中，动能算符可以表示为位置波函数的二阶导数，即使没有量子力学背景，也能理解其代表的物理意义。
• ⚪️ 高斯波函数的峰值处，周围点的平均值远小于峰值，根据海森堡不确定性原理，这意味着粒子的动量不确定性增加，从而导致动能增加。
• 🤔 视频鼓励观众思考如何利用对二阶导数的理解来构建新的直觉，例如热方程或量子物理中波函数的波长与能量的关系。

Q & A

费曼在1964年康奈尔大学的讲座中提到了什么概念，让当时的本科生感到困惑？

费曼在讲座中提到了“平均在一个球上”的概念，这个概念让当时的本科生感到困惑，因为它涉及到对函数在某点附近值的平均情况的考量，以及如何通过这种方式来理解势能和量子力学中的某些现象。

在数学和物理学中，二阶导数通常用来描述什么？

在数学和物理学中，二阶导数通常用来描述函数在某点的曲率，或者用来描述一阶导数随输入变量微小变化的变化率。

费曼是如何使用‘球上的平均’概念来直观理解二阶导数的？

费曼通过考虑一维空间中某点x0周围的点（构成一个‘球’），计算这些点的函数值的平均数，然后从这个平均数中减去x0点的函数值，以此来直观理解二阶导数。这种方法可以量化函数在一点的曲率。

在量子力学中，薛定谔方程中出现的二阶导数有什么含义？

在量子力学中，薛定谔方程中的二阶导数与系统的动能有关。通过费曼的‘球上的平均’概念，我们可以将其理解为粒子的波函数在空间中某点周围的平均值与该点波函数值的差，这与粒子的动能相关联。

在多变量微积分中，三维空间中的二阶导数是如何表示的？

在多变量微积分中，三维空间中的二阶导数使用拉普拉斯算子（Laplacian）来表示，它是二阶偏导数的一种特殊形式，用于描述函数在三维空间中的曲率。

如何使用费曼的‘球上的平均’概念来推导电荷分布产生的电势能函数的方程？

通过考虑负电荷区域，我们可以推断出，当我们从负电荷区域拉走一个正电荷时，由于吸引力，电荷会获得电势能。类似地，如果我们从正电荷区域拉走一个粒子，由于排斥力，电荷会失去电势能。使用这种直觉，我们可以猜测电势能函数的二阶导数与该点的电荷密度成比例，这实际上就是麦克斯韦方程之一。

量子力学中的位置波函数为高斯函数时，其动能算符的物理意义是什么？

如果位置波函数是一个高斯函数，这意味着粒子在空间中的位置被很好地局部化。根据费曼的‘球上的平均’概念，我们可以推断出在波函数的峰值处，周围的点的平均值要小得多，因此二阶导数会是一个大的负数。这意味着粒子具有相对较大的动能，这与海森堡不确定性原理相符，即粒子的位置不确定性越低，其动量的不确定性就越高。

为什么在量子力学中，动能算符不仅能测量粒子的速度，还能告诉我们不确定性原理如何影响其运动？

动能算符在量子力学中的形式是波函数的二阶导数，它不仅反映了粒子的速度，还隐含了位置和动量的不确定性关系。当波函数越局部化，即粒子的位置不确定性越低，其动量的不确定性就越高，从而粒子的动能和运动就会受到更大的影响。

在量子力学中，波函数的局部化程度和动能之间的关系是什么？

在量子力学中，波函数的局部化程度越小，即粒子的位置不确定性越低，根据海森堡不确定性原理，其动量的不确定性就越高。这种动量的不确定性可以导致粒子具有较大的动能，从而影响粒子的运动。

如何理解热方程中的二阶导数？

热方程中的二阶导数可以被理解为描述热如何在介质中扩散的量度。类似于费曼的‘球上的平均’概念，它可能表示在一点周围区域的温度平均值与该点温度的差，这与热流和能量传递有关。

在量子物理学中，为什么高能波函数倾向于有更小的波长？

在量子物理学中，波函数的波长与其能量成反比。根据德布罗意假说，一个粒子的波长与其动量成反比，而动量与动能成正比。因此，高能波函数意味着粒子具有较大的动量，从而导致更小的波长。

费曼是如何通过简单的物理直觉来解释复杂的数学和物理概念的？

费曼通过使用直观的物理模型和类比来解释复杂的数学和物理概念。例如，他使用‘球上的平均’概念来解释二阶导数，并通过这个概念来推导电荷分布产生的电势能函数的方程，以及量子力学中的动能算符。这种方法使得复杂的理论更加易于理解和可视化。

Outlines

00:00

😀 费曼的“小球平均值”概念

本段介绍了物理学家理查德·费曼在1964年康奈尔大学的一次讲座中提出的“小球平均值”概念。视频的目的是深入探讨费曼所讨论的内容，发展对数学和物理学中二阶导数的直观理解，并解释它为何出现在薛定谔方程、电磁学等领域。此外，还提到了由哈佛大学量子计划的几位物理博士生举办的量子短片竞赛，鼓励观众参与并提供了相关信息。

05:02

🧐 一维函数的二阶导数与“小球平均值”

本段通过一维函数的例子，解释了二阶导数与函数在一点的“小球平均值”之间的关系。通过泰勒级数展开，展示了如何计算在x_0点附近的函数值的平均，并与x_0点的函数值进行比较。这个过程揭示了二阶导数与函数曲率的联系，以及如何通过二阶导数来研究函数的弯曲程度。

10:07

📐 三维空间中的二阶导数与拉普拉斯算子

本段将一维函数的“小球平均值”概念扩展到三维空间，引入了多变量微积分中的拉普拉斯算子。讨论了在三维空间中如何计算围绕一点的函数值的平均，并展示了这与三维空间中的二阶导数（拉普拉斯算子）的关系。提供了一个链接，供有兴趣的观众进一步了解这一概念的详细推导。

15:12

🤔 利用二阶导数直觉推导物理方程

本段利用二阶导数的直觉来推导物理方程，特别是电势能函数与电荷密度之间的关系，即麦克斯韦方程之一。通过电势能的直观理解，提出了一个关于电势能二阶导数与电荷密度关系的假设，并验证了这一假设的正确性。此外，还讨论了量子力学中的动能算符，展示了如何通过二阶导数来理解量子粒子的动能，以及海森堡不确定性原理如何通过薛定谔方程中的二阶导数影响粒子的运动。

🌟 结语与鼓励深入思考

视频以鼓励观众利用对二阶导数的新理解来构建新的直觉，并思考如热方程和量子物理中波函数的能级与波长关系等问题作为结尾。同时，邀请观众在评论区提出问题，并承诺会尽力回答。

Keywords

💡费曼

费曼（Richard Feynman）是视频中提到的著名物理学家，以其独特的教学风格和对复杂物理概念的直观解释而闻名。视频中引用了费曼在1964年康奈尔大学的一个讲座，特别是他关于“球面上的平均值”的概念，这成为了视频探讨第二导数直观理解的起点。

💡第二导数

第二导数是数学和物理学中的一个重要概念，它描述了函数在某一点的曲率或变化速率的变化。视频中通过费曼的“球面上的平均值”概念，直观地解释了第二导数，展示了它如何与函数的曲率相关联，并且出现在薛定谔方程、电磁学等领域。

💡泰勒级数

泰勒级数是数学中用于将一个在某点可导的无穷次函数在某一点的邻域内表示为一个多项式的数学工具。视频中提到泰勒级数，用于展开函数，以便更好地理解当函数在某点附近变化时，其值是如何变化的，这与第一导数和第二导数的计算有关。

💡量子物理

量子物理是研究物质和能量在微观尺度下行为的物理学分支。视频中提到了量子物理，特别是在讨论量子力学中的动能算符时，这个算符与第二导数有关。通过量子物理的视角，视频探讨了第二导数如何影响粒子的动力学行为。

💡薛定谔方程

薛定谔方程是量子力学中描述量子态随时间演化的基本方程。视频中提到，第二导数在薛定谔方程中出现，这表明了它在量子物理中的重要性。通过这个方程，可以计算出量子系统的波函数，进而得到系统的能量和其他物理量。

💡动能算符

动能算符是量子力学中用于表示粒子动能的数学算符。在视频中，动能算符与第二导数联系起来，说明了它如何通过波函数的第二导数来计算粒子的动能。这个概念帮助解释了量子力学中粒子的动力学行为。

💡不确定性原理

海森堡不确定性原理是量子力学的一个基本原理，它指出粒子的位置和动量不能同时被准确知晓。视频中通过讨论动能算符和波函数的局部化，揭示了不确定性原理如何影响粒子的动力学行为，特别是动能的计算。

💡拉普拉斯算子

拉普拉斯算子是多变量微积分中的一个算符，用于计算函数在三维空间中的第二导数。视频中提到，拉普拉斯算子与三维空间中的曲率有关，并且是描述电势场中电荷分布产生的能量函数的关键。

💡麦克斯韦方程组

麦克斯韦方程组是描述电磁场和电磁力的一组基本方程。视频中通过直观的方法推导出了与电荷密度相关的电势方程，这个方程实际上是麦克斯韦方程组的一部分。这展示了如何通过直观理解来推导复杂的物理方程。

💡热方程

热方程是描述热量如何在物体内部传播的偏微分方程。虽然视频中没有直接提到热方程，但它是与第二导数相关的一个重要概念，因为它涉及到温度分布随时间的变化，这与函数的第二导数有关。

💡波函数

在量子力学中，波函数是描述粒子在空间中的概率幅度分布的数学函数。视频中讨论了波函数的局部化如何影响粒子的动能，这是通过动能算符和第二导数来解释的。波函数的概念是理解量子力学中粒子行为的关键。

Highlights

费曼在1964年康奈尔大学的讲座中提出了“小球上的平均值”概念，为理解数学和物理学中的二阶导数提供了深刻的直观理解。

视频旨在深入探讨费曼所讲述的内容，发展对薛定谔方程、电磁学等领域中二阶导数出现原因的深入理解。

作者假设观众对泰勒级数有一定的了解，即能够将函数展开为级数形式。

介绍了由哈佛量子计划举办的“量子短片大赛”，鼓励任何人参与并创作关于量子物理主题的短片。

大赛提供了赢得哈佛纪念品和参观哈佛量子研究设施的机会。

强调了一阶导数的直观含义，即函数随变量微小变化的变化率。

提出了对二阶导数的直观理解，即它告诉我们函数在邻域内的平均变化情况。

通过费曼的“小球上的平均值”概念，探讨了一维情况下函数在特定点附近点的平均值。

使用泰勒级数展开来量化函数在特定点附近点的平均值与该点函数值的差异。

通过极限过程，展示了二阶导数与函数在一点的值及其周围点的平均值之间的关系。

解释了二阶导数在几何上如何与函数的曲率相关联，以及它如何用于研究函数的凹凸性。

将一维的“小球上的平均值”概念推广到三维，引入了拉普拉斯算子，并解释了其与三维空间中二阶导数的关系。

利用新获得的直觉，直观推导出了麦克斯韦方程之一，即电势的二阶导数与电荷密度的关系。

探讨了量子力学中的动能算符，即位置表象下的二阶导数，以及它如何代表粒子的动能。

通过高斯波函数的例子，说明了量子力学中粒子的动能与其波函数的局部化程度有关。

揭示了海森堡不确定性原理如何通过量子力学中的动能算符（二阶导数）影响粒子的运动和能量。

鼓励观众利用对二阶导数的理解来构建新的直觉，例如理解热方程或量子物理中高能波函数的波长。

视频以鼓励观众思考和应用新获得的直觉，以及对量子物理的深入理解作为结尾。

Transcripts

…says the following: you don’t have to know what’s going on anywhere outside of

a little ball; if you want to know what the potential is here, you tell me what it is

on the surface of any ball, no matter how small—you don’t have to look outside,

you just tell me what it is in the neighborhood—and how much mass there is in the ball. The rule

is this…

The man you just saw speaking is physicist Richard Feynman, giving a lecture at Cornell

in 1964. I remember watching this in my dorm room my freshman year of undergrad, and I

was bewildered by Feynman’s “average on a ball” concept, whatever that meant. In

this video, I want to really dive into what Feynman is talking about, and in the process,

we’ll develop a deep intuitive understanding of what the second derivative really does

in math and physics, and why it shows up in the schrodinger equation, E&M, and elsewhere.

As a heads up, I will assume you have some familiarity with Taylor series, in so far

as you know that we can expand a function as such.

Now before we dive in, really quick I want to let you all know of a really cool opportunity

from a few Harvard physics PhD students working with the Harvard Quantum Initiative. In celebration

of World Quantum Day, they are hosting a Quantum Shorts Contest over on their HQI Blog. This

is a contest open to absolutely anyone, regardless of your background or experience in physics.

Basically, they invite you to create a short video on a quantum topic of your choosing,

using your creativity to explain some aspect of quantum physics. After submitting your

entry, you have the chance to win Harvard merchandise and even a trip to Harvard to

explore their quantum research facilities and meet the scientists that push quantum

research forward. This is a really neat opportunity hosted by some really passionate people, so

check out their blog and contest if you’re interested, I’ll have it linked in the description.

The deadline is May 14, 2024, so good luck if you choose to submit anything!

Now, to begin our journey on the second derivative, I think we should quickly review our intuition

of the first derivative. Essentially, say we have some variable x, and a point x0. Likewise,

let’s say we have some function f(x), where I’ve indicated where f of x0 lands on this

number line. If we move x a tiny bit away from x_0, we will correspondingly move f(x)

a tiny bit away from f(x0). If then we take the change in f(x) and divide by the change

in x, you intuitively get the first derivative at the point x0. Formally you’ve got all

the limits and whatnot, but this the intuition. So the first derivative intuitively tells

us how much f changes when we change x by a tiny amount.

So what about the second derivative? What does it tell us? Well, usually we’re taught

that it tells us how the first derivative changes when we change the input by a tiny

amount. But…this understanding kind of sucks. I don’t wanna know what the second derivative

tells me about the first derivative, I wanna know what it tells me about the function itself!

So…how do we go about trying to intuitively understand the second derivative? Well, here

is where we are going to follow Feynman’s lead – so let’s dig into this “average

on a ball” concept he was talking about, first in one dimension.

Let’s say we have some function, and let’s look at a particular point x_0. What I want

to do is look at the points right next to x_0, both a distance dx away. Note that this

is what a “ball” is in one dimension – it’s all the points of radius dx away from x0.

Now, again trusting Feynman for a moment, I want to know if the value of f at the points

next to x_0 are on average greater than or less than the value of f at x_0. Here we see

they are both greater, but how do we quantify this?

One way to measure this is by calculating the average of f for the points around x0

(where I’ve used this fancy double bracket to indicate the average), then subtract the

value of f at x_0. This should tell just how much higher or lower, on average, the points

around x_0 are. Take a second to make sure you understand what this quantity represents.

This might seem like a random expression, but let’s follow through with it for a moment.

First, let’s calculate the value of this average term.

The average of the two values around x0 is calculated exactly as we’d expect: by adding

then dividing by two. Now, remember that dx is supposed to be pretty small, so that should

inspire us to taylor expand both of these quantities about the point x_0.

The taylor series of the point to the right of x_0 can be written as follows, while the

series of the point to the left of x_0 can be written similarly. Now if we add the two,

note that the terms with an odd power of dx will cancel out, leaving only the terms with

an even power of dx. So, we get the following. Now note that using a first order expansion

for both terms wouldn’t have worked here. Usually that does the trick, but notice that

the first order approximation canceled out! – we’ll have more on this in a moment,

but keep this in mind. So, dividing by 2, we get that the average of the points around

x_0 can be written as follows.

Now, we can subtract the point at the center, f(x0), from both sides. To proceed, I’m

going to divide both sides by dx^2. Now, let’s take the limit as dx goes to zero on both

sides. Note that all the terms with dx^2 and higher on the right hand side will go to zero.

If we then move the ½ in front of the second derivative to the left hand side, we are left

with the following. This is a really neat result: we have found that the second derivative

at a point is related to the average of the values around that point, minus the value

of the function. And if we take a moment to think about this result, this should make

a lot of sense.

Remember that we usually use the second derivative to study the curvature of functions. If a

function is concave up, then at any point, the values of the function around that point

are on average higher, so the limit we derived a few moments ago would be positive, giving

us a positive second derivative. And if a function is concave down, then at any point,

the values of the function around that point are on average lower, so the limit is negative,

and we get a negative second derivative. So this whole “average on a ball” business

is really just a way to quantitatively capture the curvature of a function, which happens

to be related to the second derivative.

Now, what is the curvature of a straight line? Zero! Which explains why the first order terms

in the taylor expansions canceled out! Those terms represent the linear part of the function,

which contribute nothing to the curvature.

So we see that the second derivative for a single variable function has a really neat

geometric interpretation in terms of the average value around a point, minus the value at the

point itself. Now, say we wanted to extend this idea to three dimensions, how would we

do that? Necessarily, this becomes a problem in multivariable calculus, but we can guess

what the solution would be in this case.

In three dimensions, to find the average value of a function around a point, we would look

at a tiny sphere of radius dx around that point, and take the average value of all the

points on that sphere. Then, we would just subtract that average by the value in the

middle of the sphere. Our claim is that this is related to some three dimensional version

of the second derivative. It turns out, this is one hundred percent correct! In three dimensions,

the corresponding expression is as follows, where the second derivative in three dimensions

is written using this symbol here, called the Laplacian. The only difference is that

the 2 has become a 6 (and in fact, that number is always 2 times the number of dimensions

you’re in).

For those of you who have taken a multivariable calculus course, you will recognize the laplacian

and know how to calculate it, but for anyone who hasn’t, let’s just take this to mean

a second derivative in 3 dimensions, which it really is. Now, for those of you interested

in a derivation of this fact, which you are much entitled to, I’ve included a link to

a clean derivation in the description. This is one of those expressions that is much better

suited to being derived on paper where you can see each calculational step – as opposed

to me flashing a hundred equations on a screen for 20 minutes. That being said, the intuition

for this equation is exactly the same as the one dimensional case.

Now, taking this expression as a fact, we can actually already use it to intuitively

derive some of the most important equations in modern physics, without doing any tedious

math.

For example, say we have some electrical charge distribution defined by a function rho(x).

How would we come up with an equation for the potential energy function generated by

this charge distribution? Although this seems like a tough problem, we can use our newfound

intuition to give it a shot.

Let’s say we have a region of negative charge. If we take a positive charge and pull it away

from the region of negative charge, the attractive force means we have to put in work to do so,

so it gains electrical potential energy as we pull it away from this region. Similar

to how lifting something up gives it more gravitational potential energy.

Likewise, if we instead had a region of positive charge, and we pulled our particle away from

this positive region, the positive charge is being repelled, so it loses potential energy.

So, let’s use this intuition to form a differential equation! Say we have our potential energy

function defined in all space, and let’s look at some point x0. We can then examine

the average potential energy on a tiny sphere around x0. If the potential energy is bigger

at points away from x0, then our intuition tells us that there should be some negative

charge here, because this means that our particle gains potential energy by moving away from

x0.

Likewise, if the potential is smaller at points away from x0, then our intuition instead states

that there is some positive charge here.

So, summarizing all this, we can use our physicists intuition to guess that maybe an equation

of the following form is right: the second derivative of the potential (which tells us

the average of how much greater the potential is around any given point) should be proportional

to the negative of the charge density at that point x0. Take a second and digest this, and

you’ll see that this exactly describes the conclusions we made a moment ago.

And it turns out, this is exactly right, this is in fact one of Maxwell’s equations, with

A equal to one over the permittivity of free space. So we have intuitively derived an equation

without needing any fancy electromagnetic theory. Although we got somewhat lucky that

the relationship on rho wasn’t something more complex, you’d be surprised how often

nature seems to choose the simplest expression.

Now, given that this channel has been dedicated to quantum mechanics in the past, we can also

use our newfound understanding to develop further intuition into quantum phenomena.

Specifically, I want to look at the kinetic energy operator in quantum mechanics, which

in the position basis can be written as the second derivative. Now, even if you’ve never

seen this before or never even taken a course in quantum mechanics, we can still understand

why this quantity should represent kinetic energy.

To recap some quantum physics, the wavefunction is a function that tells us the probability

amplitude of where our particle is. With just this, we can use our second derivative intuition

to derive some quantum facts. First, let’s assume the position wavefunction of our particle

looks like a simple gaussian. For now, let’s assume it’s a fairly tight gaussian, meaning

our particle is well localized around some point in space.

So, how do we interpret the kinetic energy operator on this wavefunction? Well, if we

take this relation as fact for the moment, then at the peak of our wavefunction, all

the points around it are on average much much smaller, so, using our newly developed intuition,

we expect the second derivative to be a big negative number, which when multiplied by

the negative, means that this quantity is a relatively large positive number. I say

relatively because hbar is tiny, but the smaller we localize the gaussian, the bigger we make

this number.

Now, this should be a somewhat shocking result. Although it’s a bit wishy washy to interpret

an operator at a single point, this statement approximately holds true in the region where

our particle is localized. So, why should this be surprising? Well, all we did was define

where our particle was localized in space, and nowhere did we input how it was moving

or in what direction. So…why and where is our particle getting this magnitude of kinetic

energy?

What we are discovering here is in fact a vestige of the heisenberg uncertainty principle.

Note that our particle is really tightly confined in space, so we have a really low uncertainty

in its position. The uncertainty principle then dictates that we must be wildly uncertain

in our particle’s momentum, and therefore it can take on large magnitudes, increasing

our particle’s kinetic energy. And in fact, if we time evolve this initial quantum state,

the solution we would get be a gaussian that spreads out through time, since that extra

kinetic energy from momentum uncertainty pushes our particle outwards.

So, now we can see that the kinetic energy operator in quantum mechanics not only measures

how fast our particle is moving, but it also carries information on how the uncertainty

principle affects its motion: the more localized our wavefunction is, and therefore the smaller

the average values around it are, the more its motion and energy will be warped by momentum

uncertainty. I think this is absolutely fascinating, and it gives us some intuition into how the

uncertainty principle is baked into the schrodinger equation through the second derivative.

Now before wrapping up the video, I encourage you to think of how you can use this understanding

of the second derivative to build new intuitions. For example, think about what the differential

heat equation is actually saying. Likewise, in quantum physics, higher energy wavefunctions

tend to have smaller and smaller wavelengths– how can we now understand this?

I will leave this up to you to think about! As physicists and mathematicians, hopefully

I’ve given you another tool that you can use to understand our world, much in the same

way the Feynman did for me many years ago. As always, if you have any questions, feel

free to leave a comment and I’ll do my best to answer it. Hope you all had a good quantum

day!