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The ket $ | \psi(t) \rangle$ denotes the state of the system at the time $t$, and $|\psi(0) \rangle$ is nothing but the state of the system at the time $t=0$.

The probability amplitude of the state $|\psi(t)\rangle$ being in some state $| x \rangle$ is $\langle x | \psi(t) \rangle$, which when $x$ represents the spatial position of a particle is nothing but the wave function $\psi(x,t)$ (see e.g. this Phys.SE question).

Regarding the relation between probability and probability amplitude, see for example this and this Phys.SE questions.

The ket $ | \psi(t) \rangle$ denotes the state of the system at the time $t$, and $|\psi(0) \rangle$ is nothing but the state of the system at the time $t=0$.

The probability amplitude of the state $|\psi(t)\rangle$ being in some state $| x \rangle$ is $\langle x | \psi(t) \rangle$, which when $x$ represents the spatial position of a particle is nothing but the wave function $\psi(x,t)$ (see e.g. this Phys.SE question).

Regarding the relation between probability and probability amplitude, see for example this and this Phys.SE questions.

The ket $ | \psi(t) \rangle$ denotes the state of the system at the time $t$, and $|\psi(0) \rangle$ is nothing but the state of the system at the time $t=0$.

The probability amplitude of the state $|\psi(t)\rangle$ being in some state $| x \rangle$ is $\langle x | \psi(t) \rangle$, which when $x$ represents the spatial position of a particle is nothing but the wave function $\psi(x,t)$ (see e.g. this Phys.SE question).

Regarding the relation between probability and probability amplitude, see for example this and this Phys.SE questions.

The ket $ | \psi(t) \rangle$ denotes the state of the system at the time $t$, and $|\psi(0) \rangle$ is nothing but the state of the system at the time $t=0$.

The probability amplitude of the state $|\psi(t)\rangle$ being in some state $| x \rangle$ is $\langle x | \psi(t) \rangle$, which when $x$ represents the spatial position of a particle is nothing but the wave function $\psi(x,t)$ (see e.g. this Phys.SE question).

Regarding the relation between probability and probability amplitude, see for example this and this Phys.SE questions.

The notation $$ \tag 1 | \psi(t) \rangle$$ denotes the state of the system at the time $t$, and $|\psi(0) \rangle$ is nothing but the state of the system at the time $t=0$.

The probability amplitude of the state $|\psi(t)\rangle$ being in some state $| x \rangle$ is $\langle x | \psi(t) \rangle$, which when $x$ represents the spatial position of a particle is nothing but the wave function $\psi(x,t)$.

The ket $ | \psi(t) \rangle$ denotes the state of the system at the time $t$, and $|\psi(0) \rangle$ is nothing but the state of the system at the time $t=0$.

The probability amplitude of the state $|\psi(t)\rangle$ being in some state $| x \rangle$ is $\langle x | \psi(t) \rangle$, which when $x$ represents the spatial position of a particle is nothing but the wave function $\psi(x,t)$ (see e.g. this Phys.SE question).

Regarding the relation between probability and probability amplitude, see for example this and this Phys.SE questions.