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# 物理代写|量子力学代写Quantum mechanics代考|PHY350 The Scalar Product and Dual Correspondence

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## 物理代写|量子力学代写Quantum mechanics代考|The Scalar Product and Dual Correspondence

We postulate the existence of a metric or scalar product $g$ on $\mathcal{E}$, which is a function
$$g: \mathcal{E} \times \mathcal{E} \rightarrow \mathbb{C}$$
with certain properties. This notation means that $g$ is a function that takes two kets and produces a complex number; for example, we will write $g(|\psi\rangle,|\phi\rangle)$ for the complex number that results upon letting $g$ act on kets $|\psi\rangle$ and $|\phi\rangle$. The postulated properties of $g$ are the following. First, $g$ is linear in its second operand and antilinear in its first operand. Explicitly, this means
\begin{aligned} &g\left(|\psi\rangle, c_1\left|\phi_1\right\rangle+c_2\left|\phi_2\right\rangle\right)=c_1 g\left(|\psi\rangle,\left|\phi_1\right\rangle\right)+c_2 g\left(|\psi\rangle,\left|\phi_2\right\rangle\right), \ &g\left(c_1\left|\psi_1\right\rangle+c_2\left|\psi_2\right\rangle,|\phi\rangle\right)=c_1^* g\left(\left|\psi_1\right\rangle,|\phi\rangle\right)+c_2^* g\left(\left|\psi_2\right\rangle,|\phi\rangle\right) . \end{aligned}
(An antilinear function requires us to take the complex conjugate of the coefficients when evaluating it on linear combinations of vectors.) Next, $g$ is symmetric or Hermitian, which means
$$g(|\psi\rangle,|\phi\rangle)=g(|\phi\rangle,|\psi\rangle)^* .$$
Third, $g$ is positive definite, which means
$$g(|\psi\rangle,|\psi\rangle) \geq 0,$$
for all $|\psi\rangle$, with equality holding if and only if $|\psi\rangle=0$. Note that by property (16), the left hand side of Eq. (17) is necessarily real.

## 物理代写|量子力学代写Quantum mechanics代考|The Schwarz Inequality

We can now prove an important theorem, namely the Schwarz inequality. We are interested in this inequality for complex vector spaces, but it is also true in real vector spaces with a positive definite inner product, that is, in Euclidean spaces. There it is equivalent to the geometrical fact that the shortest distance between two points is a straight line. The Schwarz inequality is important in quantum mechanics because it is used to prove the Heisenberg uncertainty relations.
The Schwarz inequality says that
$$|\langle\psi \mid \phi\rangle|^2 \leq\langle\psi \mid \psi\rangle\langle\phi \mid \phi\rangle,$$
for all $|\psi\rangle$ and $|\phi\rangle$, with equality if and only if $|\psi\rangle$ and $|\phi\rangle$ are linearly dependent, that is, if they lie in the same ray. To prove the theorem, we set
$$|\alpha\rangle=|\psi\rangle+\lambda|\phi\rangle,$$
where $\lambda$ is a complex number, and we use Eq. (27) to write,
$$\langle\alpha \mid \alpha\rangle=\langle\psi \mid \psi\rangle+\lambda\langle\psi \mid \phi\rangle+\lambda^*\langle\phi \mid \psi\rangle+|\lambda|^2\langle\phi \mid \phi\rangle \geq 0 .$$

## 物理代写|量子力学代写Quantum mechanics代考|The Scalar Product and Dual Correspondence

$$g: \mathcal{E} \times \mathcal{E} \rightarrow \mathbb{C}$$

$$g\left(|\psi\rangle, c_1\left|\phi_1\right\rangle+c_2\left|\phi_2\right\rangle\right)=c_1 g\left(|\psi\rangle,\left|\phi_1\right\rangle\right)+c_2 g\left(|\psi\rangle,\left|\phi_2\right\rangle\right), \quad g\left(c_1\left|\psi_1\right\rangle+c_2\left|\psi_2\right\rangle,|\phi\rangle\right)=c_1^* g\left(\left|\psi_1\right\rangle,|\phi\rangle\right)+c_2^* g\left(\left|\psi_2\right\rangle,|\phi\rangle\right)$$
（反线性函数要求我们在对向量的线性组合进行评估时采用系数的复共轭。）接下来， $g$ 是对称的或 Hermitian 的，这意味着
$$g(|\psi\rangle,|\phi\rangle)=g(|\phi\rangle,|\psi\rangle)^* .$$

$$g(|\psi\rangle,|\psi\rangle) \geq 0,$$

## 物理代写|量子力学代写Quantum mechanics代考|The Schwarz Inequality

$$|\langle\psi \mid \phi\rangle|^2 \leq\langle\psi \mid \psi\rangle\langle\phi \mid \phi\rangle,$$

$$|\alpha\rangle=|\psi\rangle+\lambda|\phi\rangle,$$

$$\langle\alpha \mid \alpha\rangle=\langle\psi \mid \psi\rangle+\lambda\langle\psi \mid \phi\rangle+\lambda^*\langle\phi \mid \psi\rangle+|\lambda|^2\langle\phi \mid \phi\rangle \geq 0$$

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