Quantum OperatorsYouTube↗  IA↗(long shot)
Science & Technology  149.1K views 3K likes 54 dislikes   28 Mar 2016
Physics Videos by Eugene Khutoryansky 1M subscribers United States YouTube↗


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All particles are described by what we call a wave function.
所有粒子都被我们称之为波函数描述。
Every possible measurement we can make on a particle
我们可以对粒子进行每种可能的测量
is described by what we call a quantum operator.
我们称之为量子算子。
A quantum operator is a mathematical operation
量子算子是一种数学运算
on the wave function that produces another function.
在波函数上产生另一个函数。
Each type of measurement is associated with its own unique mathematical operation.
每种类型的测量都与其自己独特的数学运算相关联。
When a quantum operator is applied, the new function that results
当应用量子运算符时，会产生新的函数
typically has a different shape than the original wave function.
通常具有与原始波函数不同的形状。
But, in certain special cases, the new function has the exact same shape
但是，在某些特殊情况下，新功能具有完全相同的形状
as the original function, except for the fact that its amplitude
作为原始功能，除了它的振幅这一事实
has been scaled up or down by the exact same number everywhere.
已经按比例放大或缩小了到处都是相同的数字。
The amount by which the wave function’s amplitude has been scaled is exactly
波函数幅度的缩放量恰好是
equal to the value we end up observing for the measurement in question.
等于我们最终观察到的测量值。
Suppose that the measurement we are interested in
假设我们感兴趣的测量
is the particle’s position along the X axis.
是粒子沿X轴的位置。
The quantum operator associated with the position measurement is to multiply
与位置测量相关的量子算子是相乘的
the wave function at each point in space by the coordinate along the X axis.
通过沿X轴的坐标在空间中的每个点处的波函数。
Because each point has a different value for its X coordinate,
因为每个点的X坐标值不同，
every point along the wave function is multiplied by a different amount.
沿波函数的每个点乘以不同的量。
There is only one way the entire wave function can be scaled by the
整个波函数只有一种方法可以缩放
same constant amount everywhere when this operator is applied.
应用此运算符时，到处都是相同的常量。
This is if the original wave function is zero at all points in space except one.
这是因为原始波函数在空间中除了一个之外的所有点都是零。
If the wave function is zero at all points except one, then when we
如果波函数在除一个之外的所有点都为零，那么当我们
apply the operator, it looks as if the wave function
应用运算符，它看起来好像是波函数
has been multiplied by the same number everywhere.
在任何地方都乘以相同的数字。
The number which it is multiplied by is the
它乘以的数字是
value along the X axis where the amplitude is not zero.
沿X轴的值，幅度不为零。
And hence, it is this value that ends up being our observation for the position.
因此，正是这个价值最终成为我们对这个位置的观察。
Every possible wave function in existence can be thought of as a combination
存在的每个可能的波函数可以被认为是组合
of an infinite number of functions that are each zero at all points except one.
无限数量的函数，除了一个之外的所有点都是零。
If this function is multiplied by the real number “3”,
如果此函数乘以实数“3”，
the amplitude changes as shown.
振幅如图所示变化。
If the function is then multiplied by the imaginary number “i”,
如果函数再乘以虚数“i”，
the function is rotated by 90 degrees as shown.
如图所示，该功能旋转90度。
On the other hand, if the function is multiplied by negative “i”,
另一方面，如果函数乘以负“i”，
the function rotates by 90 degrees in the other direction.
该功能在另一个方向旋转90度。
Suppose we have this infinite combination of functions.
假设我们有这种无限的功能组合。
Suppose we multiply each of our original functions by a different constant.
假设我们将每个原始函数乘以不同的常数。
Each of these constants is a different complex number.
这些常数中的每一个都是不同的复数。
By choosing the right set of constants, we can transform our
通过选择正确的常量集，我们可以改变我们的
original set of functions into any possible final waveform.
将原始功能集合成任何可能的最终波形。
Note: Unlike what is shown here, each of these original wave functions must have
注意：与此处显示的不同，这些原始波函数中的每一个都必须具有
an amplitude that approaches infinity at the one place where they are not zero,
在一个不为零的地方接近无穷大的幅度，
and we multiply each of them by a constant to make them smaller, not larger.
然后我们将每个它们乘以一个常数，使它们变小，而不是更大。
However, infinite amplitudes are not possible to animate.
然而，无限幅度是不可能动画的。
When we observe the position, we say that
当我们观察这个位置时，我们会说
the wave function collapses to one of the original functions.
波函数折叠到原始函数之一。
After the wave function collapses to one of the original functions,
波函数折叠到原始函数之一后，
we obtain the measurement associated with that particular original function.
我们获得与该特定原始函数相关的测量值。
The probability that the wave function will collapse
波函数崩溃的概率
to each of the original functions is given by the
每个原始函数都由。给出
squares of the amplitudes of each of the corresponding constants.
每个相应常数的幅度的平方。
The probability that the wave function will collapse
波函数崩溃的概率
to each of the original functions is given by the
每个原始函数都由。给出
squares of the amplitudes of each of the corresponding constants.
每个相应常数的幅度的平方。
In the case of this sequence, the squares of the amplitudes
在该序列的情况下，幅度的平方
of each of these constants represent the square of the amplitude
这些常数中的每一个表示振幅的平方
of the total wave function at each of the positions.
每个位置的总波函数。
Therefore, the square of the amplitude of the total wave function at each location
因此，每个位置的总波函数的幅度的平方
determines the probability of observing the particle at that position.
确定在该位置观察粒子的概率。
The principles discussed here for measuring position can be generalized
这里讨论的用于测量位置的原理可以概括
to every type of quantum measurement.
对每种类型的量子测量。
Each possible type of measurement has its own unique
每种可能的测量类型都有其独特之处
mathematical formula for its quantum operator.
量子算子的数学公式。
Each quantum operator has its own set of wave functions that
每个量子算子都有自己的一组波函数
just end up being scaled up or down when this specific operator is applied.
在应用此特定运算符时，最终会按比例放大或缩小。
Each of the wave functions in this set is scaled by a different amount
该组中的每个波函数按不同的量进行缩放
when the quantum operator is applied.
当量子算子应用时。
The amount by which it is scaled represents
缩放的数量代表
the value of our observation for this measurement.
我们对此测量的观察值。
Each quantum operator has its own set of wave functions that
每个量子算子都有自己的一组波函数
just end up being scaled up or down when this specific operator is applied.
在应用此特定运算符时，最终会按比例放大或缩小。
Each of the wave functions in this set is scaled
该组中的每个波函数都被缩放
by a different amount when the quantum operator is applied.
应用量子算子时的量不同。
The amount by which it is scaled represents
缩放的数量代表
the value of our observation for this measurement.
我们对此测量的观察值。
If the particle is described by one of the wave functions that just end up
如果粒子是由一个刚刚结束的波函数描述的
being scaled up or down when this operator is applied,
在应用此运算符时按比例放大或缩小，
then the value for our measurement is known with 100% certainty.
然后我们100％确定地知道我们的测量值。
If the particle is described by a wave function that is not in this set
如果粒子是由不在此集合中的波函数描述的
of wave functions, then we can write the wave function as
波函数，然后我们可以写波函数
a combination of the wave functions that are in this set.
此集合中的波函数的组合。
When we perform the observation, we say that the wave function
当我们进行观察时，我们说波函数
collapses to one of the original functions in this set.
折叠到此集合中的一个原始函数。
We then obtain the measurement associated with that particular original function.
然后，我们获得与该特定原始函数相关联的测量。
Before we perform the observation, each of the original functions
在我们进行观察之前，每个原始功能
is multiplied by a constant, and added together.
乘以常数，并加在一起。
The square of the amplitudes of each constant determines
每个常数的幅度的平方确定
the probability that the wave function will collapse to that original function.
波函数将崩溃到原始函数的概率。
In this example, we are measuring the particle’s momentum in the X direction.
在这个例子中，我们测量粒子在X方向上的动量。
The quantum operator associated with this measurement is the equation shown.
与该测量相关联的量子算子是所示的等式。
Let us analyze each part of this equation one by one.
让我们逐一分析这个等式的每个部分。
This part of the equation is the
等式的这一部分是
partial derivative of the wave function with respect to X.
波函数相对于X的偏导数。
That is, if the distance in the X direction changes by a small amount,
也就是说，如果X方向上的距离变化很小，
then this part of the equation indicates how much the wave function will change.
那么等式的这一部分表明波函数将改变多少。
At each point, this change is a complex number,
在每个点上，这个变化是一个复数，
which can be represented with a vector as shown.
可以用所示的矢量表示。
This part of the equation consists of
等式的这一部分包括
multiplying by plank’s constant divided by two pi.
乘以木板的常数除以2 pi。
This just ends up multiplying the amplitudes of all our vectors
这最终会增加我们所有矢量的幅度
by the same constant amount.
以相同的恒定量。
This part of the equation is multiplying by negative “i”.
等式的这一部分乘以负“i”。
Multiplying by negative “i” rotates all our vectors by 90 degrees as shown.
乘以负“i”将所有向量旋转90度，如图所示。
Therefore, the new function described by these vectors
因此，这些向量描述了新的功能
has the exact same shape as the original wave function,
具有与原始波函数完全相同的形状，
except that its amplitude has been scaled by the exact same number everywhere.
除了它的幅度已经按照完全相同的数字缩放到处。
Note: Each red vector also needs to be moved to start along the x-axis. This part is not shown, as it would only change the amplitude of the following red waveform, not its shape.
注意：还需要移动每个红色矢量以沿x轴开始。此部分未显示，因为它只会改变后续红色波形的幅度，而不是其形状。
Therefore, the new function described by these vectors has the exact same shape
因此，这些矢量描述的新函数具有完全相同的形状
as the original wave function, except that its amplitude
作为原始波函数，除了它的幅度
has been scaled by the exact same number everywhere.
已经按照完全相同的数字缩放到处。
When the quantum operator for momentum is applied,
当应用动量的量子算子时，
the wave functions that are just scaled by the same number everywhere
波函数只是在任何地方按相同的数字缩放
are wave functions that look like this spiral.
波函数看起来像这个螺旋。
Suppose that the spiral has a shorter wavelength.
假设螺旋具有较短的波长。
In this case,
在这种情况下，
the partial derivative of the wave function with respect to X is larger.
波函数相对于X的偏导数较大。
Therefore, when we apply the quantum operator for momentum
因此，当我们将量子算子应用于动量时
to a wave function with a shorter wavelength, the observed momentum is higher.
对于具有较短波长的波函数，观察到的动量更高。
Not shown: Each red vector needs to be moved to start along the x-axis.
未显示：需要移动每个红色矢量以沿x轴开始。
Now suppose that we want to measure a particle’s energy.
现在假设我们想要测量粒子的能量。
The quantum operator associated with this measurement is the equation shown.
与该测量相关联的量子算子是所示的等式。
Let us analyze each part of this equation one by one.
让我们逐一分析这个等式的每个部分。
This part of the equation is proportional to
等式的这一部分与之成正比
the partial derivative of the wave function with respect to time.
波函数相对于时间的偏导数。
That is, if the time changes by a small amount, then this part of the equation
也就是说，如果时间变化很小，那么等式的这一部分
indicates how much the wave function will change.
表示波函数将改变多少。
This change is a complex number, which can be represented with vectors as shown.
此更改是一个复数，可以用所示的向量表示。
If we multiply these vectors by the imaginary number “i”,
如果我们将这些向量乘以虚数“i”，
all our vectors are rotated by 90 degrees.
我们所有的矢量都旋转了90度。
Note: Each orange vector also needs to be moved to start along the x-axis. This part is not shown, as it would only change the amplitude of the following orange waveform, not its shape.
注意：还需要移动每个橙色矢量以沿x轴开始。此部分未显示，因为它只会改变后续橙色波形的幅度，而不是其形状。
Therefore, the new function described by these vectors has the exact same shape
因此，这些矢量描述的新函数具有完全相同的形状
as the original wave function, except that its amplitude has been scaled
作为原始波函数，除了其幅度已经缩放
by the exact same number everywhere.
到处都是完全相同的数字。
When the quantum operator for energy is applied,
当应用能量的量子算子时，
wave functions like this one are just scaled by the same number everywhere.
像这样的波函数只是在任何地方都被相同的数字缩放。
Now suppose we have a wave function that is rotating more quickly.
现在假设我们有一个更快速旋转的波函数。
In this case, the partial derivate with respect to time is larger.
在这种情况下，关于时间的部分导数更大。
Hence the resultant function is scaled by a larger amount.
因此，所得函数按比例缩放。
This means that when we apply the quantum operator for energy to this
这意味着当我们将量子算子应用于此时
wave function that is rotating more quickly, the observed energy is higher.
波动功能旋转得更快，观察到的能量更高。
Not shown: Each orange vector needs to be moved to start along the x-axis.
未显示：需要移动每个橙色矢量以沿x轴开始。
If many wave functions with different energy levels are added together,
如果将具有不同能级的许多波函数加在一起，
then the result may look as shown.
然后结果可能如图所示。
If we try to measure the energy, this wave function will
如果我们试图测量能量，这个波函数会
collapse to a wave function such as the one of the two we just saw.
崩溃到波函数，如我们刚看到的两个中的一个。
Much more information about quantum mechanics
关于量子力学的更多信息
is available in the other videos on this channel.
可在此频道的其他视频中找到。
Please subscribe for notifications when new videos are ready.
新视频准备就绪时，请订阅通知。
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