Show that, if ψα^'=T ψα and ψβ^'=T ψβ, then (ψα^', ψβ^')=(ψα, ψ ψ)^*=(ψβ, ψα). From this, show t

Show that, if ψα^'=T ψα and ψβ^'=T ψβ, then (ψα^',...

Key Concepts

Norm Preservation

The norm of a state vector represents the total probability and must remain one for correctly normalized quantum states. Due to the antiunitary nature of the time reversal operator, the norm (which is derived from the inner product of the state vector with itself) is preserved when a state is transformed under time reversal. This invariance ensures that the probability interpretation of the quantum mechanical state remains consistent even when time is reversed.

Inner Product Invariance

The preservation of the inner product structure under transformation is essential in quantum mechanics because it guarantees that the transition probabilities between states are maintained. Specifically, even though the time reversal operator involves complex conjugation, the resulting inner product remains invariant, ensuring that the physical observables, which are derived from these inner products, do not change under time reversal.

Time Reversal Operator

In quantum mechanics, the time reversal operator represents the symmetry of reversing the direction of time. It acts on state vectors to yield the time-reversed state. Importantly, it is implemented as an antiunitary operator, meaning it not only reverses the dynamics but also involves complex conjugation when acting on state amplitudes.

Antiunitary Transformation

Antiunitary operators, like the time reversal operator, are characterized by the property that they preserve the inner product up to complex conjugation. When applied to a pair of state vectors, the inner product of the transformed states is the complex conjugate of the inner product of the original states in reversed order. This property is crucial for ensuring that probabilities (which are given by the norms of state vectors) remain invariant under time reversal.

Transcript

00:04 We are asked to prove that the norm in cn satisfies the norm laws, which are shown in exercise 72.

00:15 So these are actually usually used to determine whether or not a binary operation is in fact a norm.

00:27 So we want to show, in essence, that our norm in cn is actually a real norm.

00:35 So the first law, which is called n1, in this case, they're naming them so it must be pretty important.

00:42 So you should probably know these for the rest of the book.

00:47 First law n1 says for any vector u, the norm of u is always greater than or equal to 0, and the norm of u equals 0 if and only if u equals 0.

01:01 Okay.

01:01 So first of all, we'll just let u be some vector in cn.

01:12 Show that the norm of u is always non -negative.

01:22 Now, to prove this, first of all, let's write out what u is, so u is equal to u1 through un for some complex numbers ui.

01:40 Well, by the definition of norm, the norm of u, this is the square root of u dotted with the conjugate of u, which we see this is the same as.

02:02 The square root of the sum of the squares of the moduli of the components of u.

02:10 So this is, in other words, the modulus of u1 squared plus the modulus of u2 squared, all the way up to the modulus of u n squared.

02:24 Now, this is a power of two function underneath the radical, and the square root of a positive is also a positive.

02:38 So this inside expression is always greater than or equal to 0, and the outside expression this always outputs a number greater than or equal to 0.

02:58 It follows that for every u, the norm is in fact defined, and the magnitude, sorry, the norm is always also greater than or equal to 0.

03:10 So if proven part of n1, to prove the rest of n1, well, let's try to prove this statement by assuming, first of all, that the norm of u equals 0.

03:30 So when is this the case? it's proving if and only a statement.

03:38 Well, again, by definition of the norm, this would mean that zero would be equal to the square root of the modulus of u1 squared plus the modulus of u2 squared and so on, all the way up to the modulus of un squared.

03:53 Of course, we've seen the squared function always outputs a number greater than equal to zero.

04:00 And in fact, the square root function equals zero if and only if, the argument under the square root function, the modulus of u1 squared, plus the modulus of u2 squared, plus modulus of un squared, is also equal to zero.

04:19 And so we have here the sum of n squares has to be equal to zero.

04:25 And this is the sum of n squares of real numbers.

04:31 However, we know that for each ui, the modulus of ui squared must be greater than or equal to 0, and the modulus of ui is equal to 0.

04:46 We'll get to that later.

04:49 Since the modulus of each ui squared is greater than or equal to 0, it follows that the sum is only equal to 0 if each of the u .i.

05:02 Ui's in modulus squared must be equal to zero.

05:06 And this of course is only possible if the modulus of ui itself is zero.

05:12 And the modulus of ui is equal to zero.

05:16 Well this is actually a result from complex numbers which you've already shown.

05:20 But this is the case.

05:22 Well, this implies that ui must be the zero complex number.

05:27 Since every component is zero, it follows that u as components 0 -0 up through 0, or in other words, u is the zero vector.

05:40 So we have proven one direction for the second part of law n1.

05:46 Now, on the other hand, now let's assume that u is in fact the zero vector.

05:55 Well, then it follows that the norm of view by definition, this is the square root of the sum of the squares of the moduli of the components.

06:05 So this is the modulus of 0 squared plus the modulus of 0 squared and so on, up to the modulus of 0 squared n times.

06:16 But the modulus of 0 is simply 0 and 0 squared is simply 0.

06:20 So this is just the square root of 0 or 0.

06:23 And now we've proven the other direction for the second part of law n1.

06:28 So we've shown that law n1 holds...

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