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1 #+TITLE: Bugs in quantum mechanics
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2 #+AUTHOR: Dylan Holmes
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3 #+SETUPFILE: ../../aurellem/org/setup.org
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4 #+INCLUDE: ../../aurellem/org/level-0.org
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5
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6 #Bugs in Quantum Mechanics
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7 #Bugs in the Quantum-Mechanical Momentum Operator
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8
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9
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10 I studied quantum mechanics the same way I study most subjects\mdash{}
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11 by collecting (and squashing) bugs in my understanding. One of these
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12 bugs persisted throughout two semesters of
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13 quantum mechanics coursework until I finally found
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14 the paper
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15 [[http://arxiv.org/abs/quant-ph/0103153][/Self-adjoint extensions of operators and the teaching of quantum
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16 mechanics/]], which helped me stamp out the bug entirely. I decided to
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17 write an article about the problem and its solution for a number of reasons:
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18
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19 - Although the paper was not unreasonably dense, it was written for
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20 teachers. I wanted to write an article for students.
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21 - I wanted to popularize the problem and its solution because other
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22 explanations are currently too hard to find. (Even Shankar's
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23 excellent [[http://books.google.com/books/about/Principles_of_quantum_mechanics.html?id=2zypV5EbKuIC][textbook]] doesn't mention it.)
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24 - I wanted to check that the bug was indeed entirely
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25 eradicated. Attempting an explanation is my way of making
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26 sure.
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27
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28 * COMMENT
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29 I recommend the
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30 paper not only for students who are learning
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31 quantum mechanics, but especially for teachers interested in debugging
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32 them.
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33
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34 * COMMENT
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35 On my first exam in quantum mechanics, my professor asked us to
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36 describe how certain measurements would affect a particle in a
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37 box. Many of these measurement questions required routine application
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38 of skills we had recently learned\mdash{}first, you recall (or
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39 calculate) the eigenstates of the quantity
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40 to be measured; second, you write the given state as a linear
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41 sum of these eigenstates\mdash{} the coefficients on each term give
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42 the probability amplitude.
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43
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44
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45 * What I thought I knew
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46
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47 The following is a list of things I thought were true of quantum
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48 mechanics; the catch is that the list contradicts itself.
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49
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50 - For any hermitian operator: Eigenstates with different eigenvalues are orthogonal.
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51 - For any hermitian operator: Any physically allowed state can be
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52 written as a linear sum of eigenstates of the operator.
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53 - The momentum operator and energy operator are hermitian, because
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54 momentum and energy are measureable quantities.
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55 - In vacuum,
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56 - the momentum operator has an eigenstate
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57 \(p(x)=\exp{(ipx/\hbar)}\) for each value of $p$.
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58 - the energy operator has an eigenstate \(|E\rangle =
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59 \alpha|p\rangle + \beta|-p\rangle\) for any \(\alpha,\beta\) and
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60 the particular choice of momentum $p=\sqrt{2mE}$.
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61 - In the infinitely deep potential well,
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62 - the momentum operator has eigenstates with the same form $p(x) =
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63 \exp{(ipx/\hbar)}$, but because of the boundary conditions on the
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64 well, the following modifications are required.
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65 - The wavefunction must be zero everywhere outside the well. That
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66 is, \(p(x) = \begin{cases}\exp{(ipx/\hbar)},& 0\lt{}x\lt{}a;
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67 \\0, & \text{for }x<0\text{ or }x>a \\ \end{cases}\)
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68 #0,&\text{for }x\lt{}0\text{ or }x\gt{}a\end{cases}\)
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69 - no longer has an eigenstate for each value
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70 of $p$. Instead, only values of $p$ that are integer multiples of
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71 $\pi a/\hbar$ are physically realistic.
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72
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73
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74
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75 * COMMENT:
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76
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77 ** Eigenstates with different eigenvalues are orthogonal
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78
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79 #+begin_quote
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80 *Theorem:* Eigenstates with different eigenvalues are orthogonal.
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81 #+end_quote
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82
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83 ** COMMENT :
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84 I can prove this: if $\Lambda$ is any linear operator, suppose $|a\rangle$
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85 and $|b\rangle$ are eigenstates of $\Lambda$. This means that
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86
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87
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88 \(
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89 \begin{eqnarray}
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90 \Lambda |a\rangle&=& a|a\rangle,\\
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91 \Lambda|b\rangle&=& b|b\rangle.\\
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92 \end{eqnarray}
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93 \)
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94
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95 If we take the difference of these eigenstates, we find that
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96
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97 \(
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98 \begin{eqnarray}
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99 \Lambda\;\left(|a\rangle-|b\rangle\right) &=& \Lambda |a\rangle - \Lambda |b\rangle
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100 \qquad \text{(because $\Lambda$ is linear.)}\\
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101 &=& a|a\rangle - b|b\rangle\qquad\text{(because $|a\rangle$ and
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102 $|b\rangle$ are eigenstates of $\Lambda$)}
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103 \end{eqnarray}\)
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104
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105
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106 which means that $a\neq b$.
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107
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108 ** Eigenvectors of hermitian operators span the space of solutions
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109
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110 #+begin_quote
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111 *Theorem:* If $\Omega$ is a hermitian operator, then every physically
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112 allowed state can be written as a linear sum of eigenstates of
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113 $\Omega$.
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114 #+end_quote
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115
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116
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117
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118 ** Momentum and energy are hermitian operators
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119 This ought to be true because hermitian operators correspond to
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120 observable quantities. Since we expect momentum and energy to be
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121 measureable quantities, we expect that there are hermitian operators
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122 to represent them.
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123
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124
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125 ** Momentum and energy eigenstates in vacuum
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126 An eigenstate of the momentum operator $P$ would be a state
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127 \(|p\rangle\) such that \(P|p\rangle=p|p\rangle\).
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128
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129 ** Momentum and energy eigenstates in the infinitely deep well
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130
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131
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132
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133 * Can you measure momentum in the infinite square well?
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134
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135
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136
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137 ** COMMENT Momentum eigenstates
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138
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139 In free space, the Hamiltonian is \(H=\frac{1}{2m}P^2\) and the
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140 momentum operator $P$ has eigenstates \(p(x) = \exp{(-ipx/\hbar)}\).
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141
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142 In the infinitely deep potential well, the Hamiltonian is the same but
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143 there is a new condition in order for states to qualify as physically
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144 allowed: the states must not exist anywhere outside of well, as it
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145 takes an infinite amount of energy to do so.
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146
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147 Notice that the momentum eigenstates defined above do /not/ satisfy
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148 this condition.
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149
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150
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151
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152 * COMMENT
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153 For each physical system, there is a Schr\ouml{}dinger equation that
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154 describes how a particle's state $|\psi\rangle$ will change over
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155 time.
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156
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157 \(\begin{eqnarray}
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158 i\hbar \frac{\partial}{\partial t}|\psi\rangle &=&
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159 H |\psi\rangle \equiv\frac{P^2}{2m}|\psi\rangle+V|\psi\rangle \end{eqnarray}\)
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160
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161 This is a differential equation; each solution to the
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162 Schr\ouml{}dinger equation is a state that is physically allowed for
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163 our particle. Here, physically allowed states are
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164 those that change in physically allowed ways. However, like any differential
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165 equation, the Schr\ouml{}dinger equation can be accompanied by
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166 /boundary conditions/\mdash{}conditions that further restrict which
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167 states qualify as physically allowed.
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168
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169
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170
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171
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172 ** Eigenstates of momentum
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173
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174
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175
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176
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177 #In the infinitely deep well potential $V(x)=0$, the Schr\ouml{}dinger
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178
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179 #\(i\hbar\frac{\partial}{\partial t}|\psi\rangle = H|\psi\rangle\)
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180
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181
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182
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183
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184
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185
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186
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187 * COMMENT
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188
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189 #* The infinite square well potential
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190
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191 A particle exists in a potential that is
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192 infinite everywhere except for a region of length \(a\), where the potential is zero. This means that the
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193 particle exists in a potential[fn:coords][fn:infinity]
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194
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195
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196 \(V(x)=\begin{cases}0,&\text{for }\;0< x< a;\\\infty,&\text{for
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197 }\;x<0\text{ or }x>a.\end{cases}\)
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198
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199 The Schr\ouml{}dinger equation describes how the particle's state
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200 \(|\psi\rangle\) will change over time in this system.
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201
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202 \(\begin{eqnarray}
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203 i\hbar \frac{\partial}{\partial t}|\psi\rangle &=&
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204 H |\psi\rangle \equiv\frac{P^2}{2m}|\psi\rangle+V|\psi\rangle \end{eqnarray}\)
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205
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206 This is a differential equation; each solution to the
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207 Schr\ouml{}dinger equation is a state that is physically allowed for
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208 our particle. Here, physically allowed states are
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209 those that change in physically allowed ways. However, like any differential
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210 equation, the Schr\ouml{}dinger equation can be accompanied by
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211 /boundary conditions/\mdash{}conditions that further restrict which
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212 states qualify as physically allowed.
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213
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214
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215 Whenever possible, physicists impose these boundary conditions:
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216 - A physically allowed state ought to be a /smoothly-varying function of position./ This means
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217 that if a particle in the state is likely to be /at/ a particular location,
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218 it is also likely to be /near/ that location.
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219
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220 These boundary conditions imply that for the square well potential in
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221 this problem,
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222
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223 - Physically allowed states must be totally confined to the well,
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224 because it takes an infinite amount of energy to exist anywhere
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225 outside of the well (and physically allowed states ought to have
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226 only finite energy).
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227 - Physically allowed states must be increasingly unlikely to find very
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228 close to the walls of the well. This is because of two conditions: the above
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229 condition says that the particle is /impossible/ to find
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230 outside of the well, and the smoothly-varying condition says
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231 that if a particle is impossible to find at a particular location,
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232 it must be unlikely to be found nearby that location.
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233
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234 #; physically allowed states are those that change in physically
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235 #allowed ways.
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236
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237
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238 #** Boundary conditions
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239 Because the potential is infinite everywhere except within the well,
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240 a realistic particle must be confined to exist only within the
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241 well\mdash{}its wavefunction must be zero everywhere beyond the walls
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242 of the well.
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243
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244
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245 [fn:coords] I chose my coordinate system so that the well extends from
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246 \(0<x<a\). Others choose a coordinate system so that the well extends from
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247 \(-\frac{a}{2}<x<\frac{a}{2}\). Although both coordinate systems describe the same physical
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248 situation, they give different-looking answers.
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249
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250 [fn:infinity] Of course, infinite potentials are not
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251 realistic. Instead, they are useful approximations to finite
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252 potentials when \ldquo{}infinity\rdquo{} and \ldquo{}the actual height
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253 of the well\rdquo{} are close enough for your own practical
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254 purposes. Having introduced a physical impossibility into the problem
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255 already, we don't expect to get physically realistic solutions; we
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256 just expect to get mathematically consistent ones. The forthcoming
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257 trouble is that we don't.
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