Chapter 8 Vector Spaces in Quantum Mechanics
Chapter 8 Vector Spaces in Quantum Mechanics
Chapter 8 Vector Spaces in Quantum Mechanics
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<strong>Chapter</strong> 8 <strong>Vector</strong> <strong>Spaces</strong> <strong>in</strong> <strong>Quantum</strong> <strong>Mechanics</strong> 85To further simplify th<strong>in</strong>gs, we will assume that θ I = 0, i.e. the ˆl vector is <strong>in</strong> the z direction.In addition, we will make the notational simplification already used above def<strong>in</strong>ed by|S i = 1 〉 →|S 〉2|S I = ± 1 〉 →|±〉2〈S I = ± 1|S 2 i = 1 〉 →〈±|S 〉2so that |±〉 = |S z = ± 1 〉, and Eq. (8.36) becomes2or, us<strong>in</strong>g Eq. (8.37)|S 〉 = |+〉〈+|S 〉 + |−〉〈−|S 〉. (8.39)|S 〉 = cos( 1 2 θ i)|+〉 + s<strong>in</strong>( 1 2 θ i)|−〉. (8.40)We are now at the po<strong>in</strong>t at which we can beg<strong>in</strong> to supply an <strong>in</strong>terpretation to this equation.What this equation is say<strong>in</strong>g is that the comb<strong>in</strong>ation cos( 1θ 2 i)|+〉+s<strong>in</strong>( 1θ 2 i)|−〉, and |S 〉, bothrepresent the same th<strong>in</strong>g – the atomic sp<strong>in</strong> is <strong>in</strong> a state for which S i = 1 . In other words,2if we were presented with the comb<strong>in</strong>ation:1√2|+〉 + 1 √2|−〉 (8.41)we immediately see that cos( 1 2 θ i) = 1/ √ 2 and s<strong>in</strong>( 1 2 θ i) = 1/ √ 2, and hence θ i = 90 ◦ . Thusthe magnetic field is po<strong>in</strong>t<strong>in</strong>g <strong>in</strong> the direction 90 ◦ to the z direction, i.e. <strong>in</strong> the x direction,and hence the sp<strong>in</strong> state of the atom is the state |S 〉 = |S x = 1 2 〉.But what if we were presented with the comb<strong>in</strong>ation 2|+〉 + 2|−〉? Here, we cannot f<strong>in</strong>dany angle θ i , so it appears that this comb<strong>in</strong>ation is not a possible state of the atomic sp<strong>in</strong>.But we can write this as2 √ 1√22[|+〉 + √ 1 ]|−〉(8.42)2which we can now understand as represent<strong>in</strong>g 2 √ 2|S x = 1〉. Is 2 √ 2|S2 x = 1 〉 a different2physical state of the system to |S x = 1 〉? Well, it is our notation, so we can say what we2like, and what turns out to be preferable is to say that α|S 〉 describes the same physicalstate as |S 〉, for any value of the constant α. Thus, we can say that 2|+〉+2|−〉 is also a stateof the system, namely 2 √ 2|S x = 1 〉, which represents the same physical <strong>in</strong>formation2about the state of the system as |S x = 1〉.2Thus any comb<strong>in</strong>ation C + |+〉 + C − |−〉 where C ± are real numbers will always representsome state of the system, <strong>in</strong> general given by√C+ 2 + C− 2 |S i = 1 〉 (8.43)2whereS i = S · ˆn (8.44)and where ˆn is a unit vector <strong>in</strong> the direction def<strong>in</strong>ed by the angleθ i = 2 tan −1 (C−C +). (8.45)