1
00:00:00,060 --> 00:00:02,280
so when people first showed that matter
2
00:00:02,280 --> 00:00:04,620
particles like electrons can have
3
00:00:04,620 --> 00:00:06,839
wavelengths and when dubrow's showed
4
00:00:06,839 --> 00:00:09,210
that the wavelength is Planck's constant
5
00:00:09,210 --> 00:00:11,820
over the momentum people were like cool
6
00:00:11,820 --> 00:00:14,309
it's pretty sweet but you know someone
7
00:00:14,309 --> 00:00:15,389
was like wait a minute
8
00:00:15,389 --> 00:00:18,210
if this particle has wave-like
9
00:00:18,210 --> 00:00:21,300
properties and it has a wavelength what
10
00:00:21,300 --> 00:00:24,510
exactly is waving what is this wave
11
00:00:24,510 --> 00:00:26,760
we're even talking about conceptually
12
00:00:26,760 --> 00:00:29,130
it's a little strange I mean a water
13
00:00:29,130 --> 00:00:31,500
wave we know what that is it's a bunch
14
00:00:31,500 --> 00:00:33,899
of water that's oscillating up and down
15
00:00:33,899 --> 00:00:36,510
a wave on a string we know what that is
16
00:00:36,510 --> 00:00:38,790
the string itself is moving up and down
17
00:00:38,790 --> 00:00:41,100
and it extends through space but it's
18
00:00:41,100 --> 00:00:43,559
hard to imagine how is this electron
19
00:00:43,559 --> 00:00:46,530
having a wavelength and what is the
20
00:00:46,530 --> 00:00:49,140
actual wave itself so physicists were
21
00:00:49,140 --> 00:00:51,120
grappling with this issue trying to
22
00:00:51,120 --> 00:00:53,730
conceptually understand how to describe
23
00:00:53,730 --> 00:00:56,399
the wave of the electron they wanted to
24
00:00:56,399 --> 00:00:58,739
do two things they wanted a mathematical
25
00:00:58,739 --> 00:01:00,840
description for the shape of that wave
26
00:01:00,840 --> 00:01:03,180
and that's called the wave function so
27
00:01:03,180 --> 00:01:04,949
this wave function gives you a
28
00:01:04,949 --> 00:01:07,200
mathematical description for what the
29
00:01:07,200 --> 00:01:08,869
shape of the wave is so different
30
00:01:08,869 --> 00:01:11,010
electron systems are going to have
31
00:01:11,010 --> 00:01:13,290
different wave functions and this is
32
00:01:13,290 --> 00:01:16,350
this is psy it's the symbol for the wave
33
00:01:16,350 --> 00:01:19,740
function so this is psy the size symbol
34
00:01:19,740 --> 00:01:22,590
it's a function of X so at different
35
00:01:22,590 --> 00:01:25,170
points in X it may have a large value
36
00:01:25,170 --> 00:01:27,780
and may have a small value this function
37
00:01:27,780 --> 00:01:29,729
would give you the mathematical shape of
38
00:01:29,729 --> 00:01:31,439
this wave so I was one of the things
39
00:01:31,439 --> 00:01:33,090
they were trying to determine but they
40
00:01:33,090 --> 00:01:35,220
also wanted to interpret it like what is
41
00:01:35,220 --> 00:01:37,740
this wave function even mean so we've
42
00:01:37,740 --> 00:01:40,079
got two problems we want a mathematical
43
00:01:40,079 --> 00:01:41,909
description of the wave and we want to
44
00:01:41,909 --> 00:01:43,890
interpret what is this wave even mean
45
00:01:43,890 --> 00:01:46,619
now the person that gave us the
46
00:01:46,619 --> 00:01:49,140
mathematical description of this wave
47
00:01:49,140 --> 00:01:52,170
function was Irwin Schrodinger so
48
00:01:52,170 --> 00:01:54,590
Schrodinger is this guy right here
49
00:01:54,590 --> 00:01:56,189
Schrodinger's right here
50
00:01:56,189 --> 00:01:58,439
he wrote down Schrodinger's equation and
51
00:01:58,439 --> 00:01:58,950
his
52
00:01:58,950 --> 00:02:01,469
aim now is basically synonymous with
53
00:02:01,469 --> 00:02:03,240
quantum mechanics because this is
54
00:02:03,240 --> 00:02:05,430
arguably the most important equation in
55
00:02:05,430 --> 00:02:07,500
all of quantum mechanics there's a bunch
56
00:02:07,500 --> 00:02:09,389
of partial derivatives in here and
57
00:02:09,389 --> 00:02:11,849
Planck's constants but the important
58
00:02:11,849 --> 00:02:13,890
thing is that it's got the wave function
59
00:02:13,890 --> 00:02:15,780
in here now if you've never seen partial
60
00:02:15,780 --> 00:02:17,880
derivatives or calculus it's okay all
61
00:02:17,880 --> 00:02:19,860
you need to know for our purposes today
62
00:02:19,860 --> 00:02:22,739
in this video is that this equation is a
63
00:02:22,739 --> 00:02:25,769
way to crank out the mathematical wave
64
00:02:25,769 --> 00:02:28,230
function what is this function that
65
00:02:28,230 --> 00:02:30,660
gives us the shape of the wave as a
66
00:02:30,660 --> 00:02:32,370
function of X and you can imagine
67
00:02:32,370 --> 00:02:34,980
plotting this on some graph so once you
68
00:02:34,980 --> 00:02:37,530
solve for this sigh as a function of X
69
00:02:37,530 --> 00:02:39,810
you could plot what this looks like
70
00:02:39,810 --> 00:02:41,730
maybe it looks something like this and
71
00:02:41,730 --> 00:02:43,739
who knows it could do all kinds of stuff
72
00:02:43,739 --> 00:02:45,180
maybe it looks like that but
73
00:02:45,180 --> 00:02:47,160
Schrodinger's equation is the way you
74
00:02:47,160 --> 00:02:49,410
can get this wave function so
75
00:02:49,410 --> 00:02:51,420
Schrodinger gave us a way to get the
76
00:02:51,420 --> 00:02:53,430
mathematical wave function but we also
77
00:02:53,430 --> 00:02:55,530
wanted to interpret it what does this
78
00:02:55,530 --> 00:02:57,440
even mean to say that this wave function
79
00:02:57,440 --> 00:03:00,180
represents the electrons is still
80
00:03:00,180 --> 00:03:02,010
strange what does that mean
81
00:03:02,010 --> 00:03:03,780
Schrodinger tried to interpret it this
82
00:03:03,780 --> 00:03:05,850
way he said okay maybe maybe this
83
00:03:05,850 --> 00:03:08,639
electron really is like smudged out in
84
00:03:08,639 --> 00:03:11,310
space and it's charge is kind of
85
00:03:11,310 --> 00:03:13,489
distributed in different places
86
00:03:13,489 --> 00:03:16,590
Schrodinger wanted to interpret this
87
00:03:16,590 --> 00:03:20,370
wave function as charge density and I
88
00:03:20,370 --> 00:03:21,870
mean it's kind of a reasonable thing to
89
00:03:21,870 --> 00:03:22,350
do
90
00:03:22,350 --> 00:03:24,090
the way you get a water wave is by
91
00:03:24,090 --> 00:03:26,639
having water spread out through space so
92
00:03:26,639 --> 00:03:28,709
maybe the way you get an electron wave
93
00:03:28,709 --> 00:03:31,109
is to have the charge of the electron
94
00:03:31,109 --> 00:03:33,540
spread out through space but this
95
00:03:33,540 --> 00:03:35,579
description didn't work so well which is
96
00:03:35,579 --> 00:03:38,010
kind of strange Schrodinger invented
97
00:03:38,010 --> 00:03:39,720
this equation he came up with this
98
00:03:39,720 --> 00:03:41,819
equation but he couldn't even interpret
99
00:03:41,819 --> 00:03:44,430
what he was describing correctly it took
100
00:03:44,430 --> 00:03:46,260
someone else it took a guy named Max
101
00:03:46,260 --> 00:03:49,079
Born to give us the interpretation we go
102
00:03:49,079 --> 00:03:52,019
with now for this wave function Max Born
103
00:03:52,019 --> 00:03:54,000
said nah don't interpret it as the
104
00:03:54,000 --> 00:03:55,980
charge density what you should do is
105
00:03:55,980 --> 00:03:58,590
interpret this sy is giving you a way to
106
00:03:58,590 --> 00:04:00,870
get the probability of finding the
107
00:04:00,870 --> 00:04:03,780
electron at a given point in space so
108
00:04:03,780 --> 00:04:06,150
Max Born said this if you find your sigh
109
00:04:06,150 --> 00:04:07,560
like he said go ahead and use
110
00:04:07,560 --> 00:04:09,480
Schrodinger's equation use it
111
00:04:09,480 --> 00:04:11,780
get the sigh once you have
112
00:04:11,780 --> 00:04:14,720
what you do is you square this function
113
00:04:14,720 --> 00:04:16,970
so take the absolute value square it and
114
00:04:16,970 --> 00:04:18,440
what that's going to give you is the
115
00:04:18,440 --> 00:04:22,040
probability of finding the electron at a
116
00:04:22,040 --> 00:04:24,290
given point now technically it's the
117
00:04:24,290 --> 00:04:27,140
probability density but for our purposes
118
00:04:27,140 --> 00:04:28,760
you can pretty much just think about
119
00:04:28,760 --> 00:04:30,980
this as the probability of finding the
120
00:04:30,980 --> 00:04:33,170
electron at a given point so if this was
121
00:04:33,170 --> 00:04:35,030
our wave function in other words Max
122
00:04:35,030 --> 00:04:37,010
Born would tell us that points where
123
00:04:37,010 --> 00:04:39,020
it's zero these points right here where
124
00:04:39,020 --> 00:04:41,480
the value is zero there is a zero
125
00:04:41,480 --> 00:04:43,460
percent chance you're going to find the
126
00:04:43,460 --> 00:04:45,290
electron there points where there's a
127
00:04:45,290 --> 00:04:49,070
large value of sy be it positive or
128
00:04:49,070 --> 00:04:51,260
negative there's going to be a large
129
00:04:51,260 --> 00:04:53,600
probability of finding the electron at
130
00:04:53,600 --> 00:04:56,150
that point and we could say the odds of
131
00:04:56,150 --> 00:04:57,650
finding the electron at a given point
132
00:04:57,650 --> 00:04:59,930
here are going to be largest for this
133
00:04:59,930 --> 00:05:02,360
value of x right here because that's the
134
00:05:02,360 --> 00:05:04,640
point for which the wave function has
135
00:05:04,640 --> 00:05:07,730
the greatest magnitude but you won't
136
00:05:07,730 --> 00:05:10,220
necessarily find the electron there if
137
00:05:10,220 --> 00:05:12,800
you repeat this experiment over and over
138
00:05:12,800 --> 00:05:16,010
you may find the electron here once you
139
00:05:16,010 --> 00:05:17,720
may find it over here you may find it
140
00:05:17,720 --> 00:05:19,970
there next time you have to keep taking
141
00:05:19,970 --> 00:05:21,410
measurements and if you keep taking
142
00:05:21,410 --> 00:05:22,520
measurements you'll get this
143
00:05:22,520 --> 00:05:23,870
distribution where you find a lot of
144
00:05:23,870 --> 00:05:27,110
them here a lot of them there a lot of
145
00:05:27,110 --> 00:05:28,820
them here and a lot of them here always
146
00:05:28,820 --> 00:05:30,290
where there's these peaks you get more
147
00:05:30,290 --> 00:05:32,419
of them then you would have at other
148
00:05:32,419 --> 00:05:34,760
points where the values were smaller you
149
00:05:34,760 --> 00:05:36,530
build up a distribution that's
150
00:05:36,530 --> 00:05:39,050
represented by this wave function so the
151
00:05:39,050 --> 00:05:40,790
wave function does not tell you where
152
00:05:40,790 --> 00:05:42,650
the electron is going to be it just
153
00:05:42,650 --> 00:05:44,600
gives you the probability and
154
00:05:44,600 --> 00:05:47,000
technically the square of it gives you
155
00:05:47,000 --> 00:05:48,350
the probability of finding the electron
156
00:05:48,350 --> 00:05:50,360
somewhere so even it points down here
157
00:05:50,360 --> 00:05:52,220
with the wave function has a negative
158
00:05:52,220 --> 00:05:53,990
value I mean you can't have a negative
159
00:05:53,990 --> 00:05:57,050
probability you square that value that
160
00:05:57,050 --> 00:05:59,210
gives you the probability of finding the
161
00:05:59,210 --> 00:06:01,010
electron in that region so in other
162
00:06:01,010 --> 00:06:02,780
words let's get rid of all this let's
163
00:06:02,780 --> 00:06:06,200
say we sold some Schrodinger equation or
164
00:06:06,200 --> 00:06:08,240
we were just handed a wave function and
165
00:06:08,240 --> 00:06:09,590
we were told it looks like this and
166
00:06:09,590 --> 00:06:11,390
we're asked where are you most likely to
167
00:06:11,390 --> 00:06:13,729
find the electron well the value of the
168
00:06:13,729 --> 00:06:15,919
wave function is greatest at this point
169
00:06:15,919 --> 00:06:18,020
here so you'd be most likely to find the
170
00:06:18,020 --> 00:06:20,270
electron in this region right here you'd
171
00:06:20,270 --> 00:06:22,430
have no shot of finding it right there
172
00:06:22,430 --> 00:06:23,930
you'd have pretty good odds of finding
173
00:06:23,930 --> 00:06:25,340
it right here or
174
00:06:25,340 --> 00:06:27,110
here but you'd have the greatest chance
175
00:06:27,110 --> 00:06:29,180
of finding it in this region right here
176
00:06:29,180 --> 00:06:31,070
so you have to repeat this measurement
177
00:06:31,070 --> 00:06:33,650
many times in quantum mechanics one
178
00:06:33,650 --> 00:06:35,600
measurement doesn't verify that you've
179
00:06:35,600 --> 00:06:37,430
got the right wave function because if I
180
00:06:37,430 --> 00:06:39,169
do one experiment and measure one
181
00:06:39,169 --> 00:06:41,210
electron Boop I might find the electron
182
00:06:41,210 --> 00:06:42,410
right there that doesn't really tell me
183
00:06:42,410 --> 00:06:44,630
anything I have to repeat this over and
184
00:06:44,630 --> 00:06:47,330
over to make sure the relative frequency
185
00:06:47,330 --> 00:06:49,940
of where I'm finding electrons matches
186
00:06:49,940 --> 00:06:52,280
the wave function I'm using to model
187
00:06:52,280 --> 00:06:54,500
that electron system so that's what the
188
00:06:54,500 --> 00:06:56,479
wave function is that's what it can do
189
00:06:56,479 --> 00:06:59,000
for you although if I were you I'd still
190
00:06:59,000 --> 00:07:00,800
be unsatisfied I'd be like wait a minute
191
00:07:00,800 --> 00:07:03,290
okay that's fine and good wave function
192
00:07:03,290 --> 00:07:05,660
can give us the probability or the
193
00:07:05,660 --> 00:07:07,550
probability density of finding the
194
00:07:07,550 --> 00:07:09,229
electron in a given region but we
195
00:07:09,229 --> 00:07:11,150
haven't answered the question what is
196
00:07:11,150 --> 00:07:14,600
waving here and what exactly is this
197
00:07:14,600 --> 00:07:16,729
wave function is this a physical object
198
00:07:16,729 --> 00:07:19,389
sort of like a water wave or even an
199
00:07:19,389 --> 00:07:21,500
electromagnetic wave or is this just
200
00:07:21,500 --> 00:07:23,840
some mathematical trickery that we're
201
00:07:23,840 --> 00:07:25,600
using that has no physical
202
00:07:25,600 --> 00:07:28,100
interpretation other than giving us
203
00:07:28,100 --> 00:07:30,860
information about where the electron is
204
00:07:30,860 --> 00:07:32,900
going to be and I've got good news and
205
00:07:32,900 --> 00:07:33,830
bad news
206
00:07:33,830 --> 00:07:36,800
the bad news is that people still don't
207
00:07:36,800 --> 00:07:39,289
agree on how to interpret this wave
208
00:07:39,289 --> 00:07:41,660
function yes they know that the square
209
00:07:41,660 --> 00:07:43,220
of it gives you the probability of
210
00:07:43,220 --> 00:07:45,380
finding the electron in some region but
211
00:07:45,380 --> 00:07:47,270
people differ on how they're supposed to
212
00:07:47,270 --> 00:07:49,370
interpret it past that point for
213
00:07:49,370 --> 00:07:52,160
instance is this wave function the wave
214
00:07:52,160 --> 00:07:54,800
function of a single electron or is this
215
00:07:54,800 --> 00:07:57,020
wave function really the wave function
216
00:07:57,020 --> 00:08:00,700
of a system an ensemble of electrons all
217
00:08:00,700 --> 00:08:02,780
similarly prepared that you're going to
218
00:08:02,780 --> 00:08:05,000
do the experiment on in other words does
219
00:08:05,000 --> 00:08:07,010
it describe one electron or only
220
00:08:07,010 --> 00:08:09,710
described a system of electrons does it
221
00:08:09,710 --> 00:08:11,630
not describe the electron at all but
222
00:08:11,630 --> 00:08:14,210
only our measurement of the electron and
223
00:08:14,210 --> 00:08:16,610
what happens to this wave function when
224
00:08:16,610 --> 00:08:18,710
you actually measure the electron when
225
00:08:18,710 --> 00:08:20,210
you measure the electron you find it
226
00:08:20,210 --> 00:08:22,340
somewhere and at that moment there's no
227
00:08:22,340 --> 00:08:24,530
chance of finding it over here at all so
228
00:08:24,530 --> 00:08:26,690
does the act of measuring the electron
229
00:08:26,690 --> 00:08:29,599
cause some catastrophic collapse in this
230
00:08:29,599 --> 00:08:31,910
wave function that's not described by
231
00:08:31,910 --> 00:08:34,010
Schrodinger's equation these and many
232
00:08:34,010 --> 00:08:36,500
more questions are still debated and not
233
00:08:36,500 --> 00:08:37,820
completely understood
234
00:08:37,820 --> 00:08:38,690
that's the bad
235
00:08:38,690 --> 00:08:40,909
News the good news is that we don't
236
00:08:40,909 --> 00:08:43,190
really need to understand that to make
237
00:08:43,190 --> 00:08:46,160
progress everyone knows how to use the
238
00:08:46,160 --> 00:08:48,350
wave function to get the probabilities
239
00:08:48,350 --> 00:08:50,089
of measurements you can have your
240
00:08:50,089 --> 00:08:52,220
favorite interpretation but luckily
241
00:08:52,220 --> 00:08:54,290
pretty much regardless of how you
242
00:08:54,290 --> 00:08:56,660
interpret this wave function as long as
243
00:08:56,660 --> 00:08:58,189
you're using it correctly to get the
244
00:08:58,189 --> 00:09:00,170
probabilities of measurements you can
245
00:09:00,170 --> 00:09:02,389
continue making progress testing
246
00:09:02,389 --> 00:09:04,850
different models and correlating data to
247
00:09:04,850 --> 00:09:06,410
the measurements that people make in the
248
00:09:06,410 --> 00:09:07,870
lab now I'm not saying that
249
00:09:07,870 --> 00:09:09,980
interpretations of this wave function
250
00:09:09,980 --> 00:09:12,199
are not important people have tried
251
00:09:12,199 --> 00:09:14,360
cracking this nut for over a hundred
252
00:09:14,360 --> 00:09:16,910
years and it's resisted maybe that's
253
00:09:16,910 --> 00:09:19,100
because it's a waste of time or maybe
254
00:09:19,100 --> 00:09:20,990
it's because the difficulty of figuring
255
00:09:20,990 --> 00:09:23,329
this out is so great that whoever does
256
00:09:23,329 --> 00:09:25,189
it will go down in history as one of the
257
00:09:25,189 --> 00:09:27,560
great physicists of all time it's hard
258
00:09:27,560 --> 00:09:29,839
to tell right now but what's undebatable
259
00:09:29,839 --> 00:09:31,550
is that for about a hundred years now
260
00:09:31,550 --> 00:09:34,370
we've been able to make progress with
261
00:09:34,370 --> 00:09:36,439
quantum mechanics even though we differ
262
00:09:36,439 --> 00:09:39,110
on how exactly to interpret what this
263
00:09:39,110 --> 00:09:41,449
wavefunction really represents so
264
00:09:41,449 --> 00:09:43,310
recapping the wavefunction gives you the
265
00:09:43,310 --> 00:09:45,410
probability of finding a particle in
266
00:09:45,410 --> 00:09:48,380
that region of space specifically the
267
00:09:48,380 --> 00:09:50,480
square of the wave function gives you
268
00:09:50,480 --> 00:09:53,389
the probability density of finding a
269
00:09:53,389 --> 00:09:55,610
particle at that point in space this
270
00:09:55,610 --> 00:09:57,829
almost everyone has agreed upon whether
271
00:09:57,829 --> 00:09:59,540
the wave function has deeper
272
00:09:59,540 --> 00:10:00,410
implications
273
00:10:00,410 --> 00:10:03,170
besides this people differ but that
274
00:10:03,170 --> 00:10:04,970
hasn't yet stopped us from applying
275
00:10:04,970 --> 00:10:07,339
quantum mechanics correctly in a variety
276
00:10:07,339 --> 00:00:00,000
of different scenarios